GEOMETRY 


AND 


COLLINEATION   GROUPS 


OF  THE 


FINITE    PROJECTIVE    PLANE 

PG(2,22) 


A    DISSERTATION 

Presented  to  the  Faculty  of  Princeton  University 

in  Candidacy  for  the  Degree  of 

Doctor  of  Philosophy 

(department     of      mathematics) 


BY 


ULYSSES  GRANT  MITCHELL 


PRINCETON 
1910 


GEOMETRY 

AND 

COLLINEATION   GROUPS 

OF    THE 

FINITE   PROJECTIVE   PLANE 

PG(2,22) 


A    DISSERTATION 

Presented  to  the  Faculty  of  Princeton  University 

in  Candidacy  for  the  Degree  of 

Doctor  of  Philosophy 

(  d  b  p  a  r  t  m  b  nt      of      maths  m  a  t  i  c  s  ) 


BY 

ULYSSES  GRANT  MITCHELL 


»RINCETON  ■ 
1910 


WS 


Copies  of  this  dissertation  may  be  obtained  on  application  to  the  University 
Library,  Princeton,  N.  j.  The  price  for  each  copy  is  50  cents,  which  includes 
postage. 


ERRATA 


Page     6,  line  21,  add  -j-a.y/il:i-  to  the  left  member  of  the  equation. 

Page   ii,  line     4,  in  place  of  "PG(2,2")"  read  'TG(2,2  )." 

Page   12,  line   io,  in  place  of  'Vs'— **-f-*8"  read  "pxu'—ixt+xt" 

Page   12,  line   II,  in  place  of    "(/-J-p-f  O"  read  "  (p*-|_^-f-i) /• 

Page  1 8,  line  30,  in  place  of  "0//"  read  "or." 

Page  20,  line   23,  In  place  of  '7,*  read  UI3"  (twice). 


-  •  ■       •  *   • 

•  •• 


Press  of  The  Journal- World 

Lawrence,  Kansas 

1914 


2KK021 


GEOMETRY  AND  COLLINEATION  GROUPS  OF  THE  FINITE 
PROJECTIVE  PLANE  PG  (2,2-).* 


§1.  Definition  of  a  Finite  Projective  Plane. 

§2.  Preliminary  Theorems. 

§3.  Types  of  Collineations  in  PG  (2,2n). 

£4.  Cyclic  Groups  in  PG  (2,2-). 

§o.  The  Group  of  Determinant  Unit}' — Go0160- 

§0.  The  Group  Leaving  Invariant  an  Imaginary  Triangle — G63. 

$7.  Invariant  Real  Configurations  and  Their  Groups. 

§8.  Subgroups  of  the  Group  G2880  Which  Leaves  a  Line  Invariant. 


§/.  Definition  of  a  Finite  Projective  Plane. 

• 

The  definition  and  general  properties  of  finite  projective  spaces  together  with 
references  to  the  literature  of  the  subject  may  be  found  in  a  paper  by  veblen 
and  bussey  in  the  Transactions  of  the  American  Mathematical  Society,  Vol.  7, 
pp.  241 -259.  They  used  the  symbol  PG(k,pn),  where  k,p,n  are  integers  and  p  is 
a  prime,  to  indicate  a  finite  projective  space  of  k  dimensions  having  pn-|-  1  points 
to  the  line.  It  is  the  purpose  of  this  paper  to  discuss  some  of  the  properties  of  the 
PG(2,2")  and  to  determine  all  subgroups  of  the  group  of  projective  collineations 
in  PG(2,22). 

We  give  a  brief  summary  of  the  analytic  and  synthetic  definitions  of  a  finite 
projective  plane. 

If  xx.x.,,x.v  are  marks  of  a  Galois  fieldf  [designated  by  GF(pn)]  of 
order  pn  there  are  (p"n — l)/(p — l)=P"n-f-Pn+l  elements  of  the  form  (xt>xstjr,) 
provided  that  the  elements   (x^x.^x.^)   and   (Ix^lx^lx^)   indicate  the  same  element 


*  Presented  to  the  American  Mathematical  Society,  April  29,  1911 

+  For  definition  and  properties  of  a  Galois  field  see  E.  H.  MOORE,  Subgroups  of  the 
Generalized  Fjnite  Modular  Group,  University  of  Chicago  Dec.  Pub,  Vol.  IX.,  pp.  141- 
156;  L.  E.  DICKSON,  Linear  Groups,  pp.  1-14. 


28802 t 


2  U.  G.  Mitchell:  Geometry  and 

when  /  is  any  mark  other  than  zero  and  provided  that  (0,0,0).  be  excluded  from 
consideration.    These  elements  constitute  a  finite  projective  plane  if  the  equation 

M,  Xy  -f  -  WoX2  +  «3*3=  O 

[the  domain  for  coefficients  and  variables  being  the  GF(pn)]  be  taken  as  the  equa- 
tion of  a  line  except  when  ui=u2=u^=o.  The  line  is  denoted  by  the  symbol 
(w,,h,,w3)  and  the  symbols  (m„«2,m3)  and  (/«1,/«2,/»R)  where  /  is  any  mark  other 
than  zero  denote  the  same  line.  The  points  of  a  line  are  those  points  whose  co- 
ordinates (x  ,,*,,*,,)  satisfy  its  equation. 

Taking  0,1,  i  and  r  for  the  marks  of  the  GF  (22)  where  i  is  defined  as  a  root 
of  the  equation  r=;'+  /  and  hence  r*=/  (mod.  2)  the  PG(2,2:)  so  defined  may 
be  exhibited  in  the  table  of  alignment  given  on  the  opposite  page.  In  the  analytical 
processes  of  PG(2,2n)  no  distinction  need  be  made  between  plus  and  minus  signs 
since  -1=1   (mod.  2). 

Synthetically  a  finite  projective  plane  may  be  defined  as  a  set  of  elements  which 
for  suggestiveness  are  called  points,  arranged  in  subsets  called  lines  and  subject  to 
the  following  conditions: 

I.  The  set  contains  a  finite  number,  greater  than  one,  of  lines,  and  each  line 
contains  pn-f-i  points  (p  and  n  integers  and  p  a  prime). 

II.  If  A  and  B  are  distinct  points  there  is  one  and  only  one  line  that  contains 
A  and  B. 

III.  All  the  points  considered  are  in  the  same  plane. 

From  this  definition  it  follows*  that  the  principle  of  duality  is 
valid  in  the  plane  so  defined,  that  there  are  pn-\-l  lines  through  each  point  and 
that  the  total  number  of  points  in  the  plane  is  p2n+pn+l. 

In  a  PG  (2,22)  there  are  then  21  points  and  21  lines.  The  following  set  of  ele- 
ments arranged  in  21  lines  of  5  elements  each  will  be  seen  to  satisfy  the  given 
synthetic  definition  and  to  be  identical  with  the  table  given  opposite. 

0     1     2    3     4     5     6     7     8     9     10  11  12  13  14  15  16  17  18  19  20 
.      1     2    3    4    5     6     7     8     9     10  11  12  13  14  15  16  17  18  19  20  0 
4    5     6     7    8     9     10  11  12  13  14  15  16  17  18  19  20  G     1     2     3 
14  15  16  17  18  19  20  0     1     2     3     4     5     6     7    8     9     10  11  12'  13 
16  17  18  19  20  0     1     2     3     4    5     6     7     8    9     10  11  12  13  14  15 

§2.  Preliminary  Theorems. 

Theorem  1.  In  PG(2,2n)  the  diagonal  points  of  a  complete  quadrangle 
are  collinear. 

Proof.  Let  three  of  the  vertices,  A,  B  and  C  of  the  quadrangle  (Fig.  1)  be 
taken  as  the  triangle  of  reference  and  let  the  fourth  vertex,  D,  be  assigned  the  co- 
ordinates (1,1,1).    The  intersections  of  AB  with  CD,  AC  with  BD  and  AD  with 


*  Cf.  VEBLEN  and  YOUNG,  Projective  Geometry,  Vol.  I,  pp.  16-17. 


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4  U.  G.  Mitchell:  Geometry  and 

3C      determine      the      diagonal      points      P,Q,R      as      (1,0,1),      (i,J,o)      and 
(o,/,/)   respectively,  which  are  collinear  on  the  line  x1-\-x2-\-x3=^0. 


B(oo0 


F1G.1. 

Since  the  quadrangle  A,B,C,D  is  projectively  equivalent  to  any  other  quad- 
rangle in  the  plane  the  proof  is  complete. 

The  line  joining  the  diagonal  points  of  any  complete  quadrangle  will  be  re- 
ferred to  as  the  diagonal  line  of  the  quadrangle. 

Definition  of  conic.  A  point  conic  is  defined  as  the  locus  of  the  points  of  in- 
tersection of  corresponding  lines  in  two  projective  non-perspective  pencils  of 
lines.  A  line  conic  is  defined  as  consisting  of  the  lines  that  join  corresponding 
points  in  two  projective  non -perspective  ranges  of  points.  In  PG  (2,2n)  the 
number  of  points  in  a  point  conic  is  2n ~|-1,  the  number  of  lines  in  a  pencil,  and 
the  number  of  lines  in  a  line  conic  is  2n-j-l,  the  number  of  points  in  a  range. 
Hence  in  PG(2,22)  a  point  conic  consists  of  any  five  points  no  three  of  which 
are  collinear  and  a  line  conic  consists  of  any  five  lines  no  three  of  which  are  con- 
current. 

A  tangent  to  a  point  conic  is  defined  as  a  line  which  has  one  and  only  one 
point  in  common  with  the  conic.  There  is  one  and  only  one  tangent  at  a  given 
point  on  a  conic  since  there  are  2n-j-l  lines  through  the  point  and  2n  lines  join- 
ing it  to  other  points  of  the  conic. 

By  taking  the  equations  of  two  projective  pencils  of  lines  XP-\-/xQ=o  and 
\P'-\-lxQ'=o  [P=0,  Q*=0,  P'=o,  Q'=o  being  equations  in  abbreviated  no- 
tation of  linej  in  PG(2,L;:)  and  A  and  /x  marks  of  the  GF(2n)]  and  eliminating 
A  and  /x  it  is  readily  shown  that  the  equation  of  a  point  conic  is  a  homogen- 
eous equation  of  the  second  degree  in  three  variables  with  coefficients  in  the 
GF(2n).     Similarly,  using  line  coordinates  it    may  be  shown    that  the  equation 


Collin eation  Groups  of  PG(2,22).  5 

of  a  line  conic  is  also  a  homogeneous  equation  of  the  second  degree  in  three  vari- 
ables with  coefficients  in  the  GB  (2*). 

Theorem    2.    Every  equation  of  the  form 

i>i 

where  the  coefficients  are  marks  of  the  GF(.2n)  is  satisfied  by  the  coordinates  of 
at  least  one  point  in  PG(2,2n). 

Proof.  Suppose  neither  «,,  nor  a,2  to  be  zero.  Taking  x3=i  the  equation 
reduces    to    aux12-{-(al2x2-\-ai3)x1-{-a22x22-\-a23x2-\-a33=o    which    is    satisfied    by 

and    x2==  —        Moreover,    since    in    the    GF(2")      every     mark     satisfies     the 
ai2 

n 

equation  x2  =*  it  is  a  perfect  square  and  since  — /=/  (mod.  2)  its  square  root 
is  unique.  These  values  for  xt  and  x2  are  therefore  uniquely  determined  and 
lie  in  the  GF(2").       If  alt=0,   F(xl,x._i,x.d)=o   is  satisfied   by    (i,o,o)    and   if 

(in  ^t  o  and  alo=0  h  (x^x^.x^^o  is  satisfied  by  (  \  ~,    i1'0)- 
Theorem    3.    Every  equation  of  the  form 

1,2,3, 

F(x1,x2,xJ=     S     tfij*i*j=0  dgj,> 

i,j 
where  x,,x2,x3  are  point  coordinates  and  the  coefficients  are  marks  of  the  GF(2Ti) 
represents  a  point  conic  in  PG(2,2n). 

Proof.  Since  by  the  previous  theorem  F(x1x2,x3)=o  is  satisfied  by  at  least 
one  point  in  PG(2,2n),  by  means  of  a  linear  transformation  of  that  point  into  the 
point   (o,o, i )   F(x^,x2,x3)=o  can  be  transformed  into  the  equation 

F,  (*,,*,,,*,)    =    b11x12-{-b12xlx2  -±b22x22+bx3xxx3+b2ix?x3=0 
This  equation  may  be  written 

*i(*u*iH-*ia*a+*ia*s)    +  **(**2*iH-*i8*«)""0 

which  is  seen  to  be  the  locus  of  points  of  intersection  of  corresponding  lines  in  the 

two  projective  pencils  of  lines 

lxi+m  (*u*l-|-*ia*a+*i3*a)  =o  and  lxx-\-m  (b22x2-{-b23x3)=o 

Hence  F(xltx2,x3)=o  represents  the  locus  previously  defined  as  a  point  conic. 

Theorem    4.    In  PG(2,2n)  all  tangents  to  the  point  conic 

1,2,3, 
F(xltx2,x3)==     2     a,Jx,jf]=o  (^i) 

i.j 
are  concurrent  and  their  point   of  concurrence  is    (a23/il3,a  ,.,). 


6  U.  G.  Mitchell:  Geometry  and 

Proof.     The  line  f^-f  r2*2+f3*3==0,  the  domain  for  variables  and  coefficients 

being  the  GF(2"),  will  be  a  tangent  to  the  conic  F  (Xl,x2,xs)=0  in  case  it  has  but 

one  point  in  common  with  the  conic.     Eliminating  xt  from  the  two  equations  gives 

(a„<V  +  al2cxc2)  x22  -f  cx  (fl12f3  +  alsc2  +  ancx)    x2x3    + 

(<*11<V      +      "l3flf3  +      <*MCl)      *32=0 

The  condition  that  this  shall  be  a  perfect  square  is  (assuming  cx  not  o) 
<V»28  +  *i3f2  +  <y*i2=0  which  is  seen  to  be  the  condition  that  the  line 
H*\  +  f2*2  +  czx3T=°  Pass  through  the  point  (a23,al3,a12).  If  cx— O  either 
f2  or  r3  is  not  zero  and  x2  or  x3  can  be  eliminated. 

If  an  outside  point  of  a  conic  be  defined  as  any  point  of  intersection  of  tangents 
to  the  conic,  it  follows  that  in  PG(2,2n)  a  conic  has  but  one  outside  point.  Hence 
the  point  of  concurrence  of  tangents  to  a  conic  will  be  referred  to  as  the  outside 
point  of  the  conic. 

Corollary  I.  Every  line  through  the  outside  point  of  a  conic  is  a  tangent  to  the 
conic. 

Corollary  2.  Through  any  point  in  PG  (2,2n)  other  than  the.  outside  point  of 
a  given  conic  passes  one  and  but  one  tangent  to  the  conic. 

Corollary  J.  In  PG(2,2n)  the  condition  for  degeneracy  of  a  conic  is  that  the  co- 
ordinates of  its  outside  point  satisfy  its  equation. 

For  the  general  conic  F(jf,,Av.Ar3)=o  the  condition  is 

Corollary  4.  In  PG(2,2ri)  six  and  but  six  points  can  be  chosen  such  that  no 
three  of  the  set  are  collinear. 

For,  any  five  points  no  three  of  which  are  collinear  determine  a  conic  which  to- 
gether with  its  outside  point  constitutes  a  set  of  six  points  no  three  of  which  are 
collinear.  The  existence  of  any  other  point  not  collinear  with  any  two  of  these 
contradicts  Corollary  2  above. 

Corollary  5.  In  PG(2,22)  the  diagonal  line  of  the  complete  quadrangle  of  any 
four  points  of  a  conic  is  the  tangent  of  the  fifth  point. 

For,  in  PG(2,22)  there  are  five  points  to  the  line  and  hence  the  diagonal  line 
contains  two  points  other  than  the  diagonal  points.  These  two  points  together 
with  the  four  points  determining  the  quadrangle  form  a  set  of  six  points  no  three 
of  which  are  collinear.  It  follows,  then,  from  Corollary  4  that  the  two  points  on  the 
diagonal  line  arc  the  fifth  point  and  the  outside  point  respectively  of  any  conic 
passing  through  the  points  of  the  quadrangle. 

§  3'  Types  of  Collintations  in  PG(2,2n). 
Transformations  on  the  line.     To  determine  the  one-dimensional  transformations 
in  PG(2,2n)   it  is  sufficient  to  consider  the  transformations  of  points  on  the  line 
x3=0  and  any  point  on  it  can  be  represented  by  two  coordinates.     Hence  the  gen- 
eral projective  transformation  of  points  on  a  line  in  PG(2,2n)  may  be  written 

T:  pAr,'=fl,,Ar1-{-ff,2Af2,  {'1=1,2), 

where    the    determinant    A--=|<7,j]    of    the    transformation    is    not    zero    and    the 


Collin eation  Groups  of  PG(2,22).  7 

domain  for  variables  and  coefficients  is  the  GF(2n).  In  the  usual  man- 
ner we  put  xi'=xl,  (*— 1,2)  in  order  to  determine  the  fixed  elements  and  con- 
sider the  characteristic  equation 

/>2+("u+*22)p+A=0  C1) 

which  expresses  the  condition  for  consistency. 

According  as  (1)  has  one,  two,  or  no  roots  in  the  GF(2n)  T  has  one,  two,  or  no 
fixed  points  in  PG(2,2n)  and  is  designated  correspondingly  as  parabolic,  hyper- 
bolic ,  or  elliptic.  In  PG(  2,2")  there  are  2n  (2n-l)  equations  of  the  form  (1)  since  A  is 
not  zero  and  there  are  2n-l  marks  other  than  zero  in  the  GF(2n).  One  half  of  these 
equations  have  both  roots  in  the  GF(2n)  and  the  other  half  have  no  roots  in  the 
GF(2n).  Of  those  having  roots  in  the  GF(2n),  2n-l  have  coincident  and 
(2n-l)  (2n_1-l)  have  distinct  roots.  The  necessary  and  sufficient  condition  that 
(1)  shall  have  coincident  roots  is  that  an^Etf22(mod.2).  When  tfn==tf22  is  substi- 
tuted in  T  it  is  found  that  T2  becomes  the  identical  collineation  and  hence  every 
parabolic  transformation  in  PG(2.2n)  is  of  period  two.  An  hyperbolic  transforma- 
tion permutes  2n-l  points  and  hence  its  period  is  2n-l  or  some  factor  of  2n-l. 
Similarly  the  period  of  an  elliptic  transformation  must  be  2n— (-1  or  some  factor 
of  2n+l. 

Suppose  (1)  to  be  irreducible  in  the  GF(2n).  From  the  theory  of  Galois  fields 
we  know  that-  its  roots  then  are  marks  plt  p*n  of  the  GF(22n)  conjugate  with 
respect  to  the  GF(2n).     Substituting  these  values  in  T  we   find   the   invariant 

points  of  T  to  be  («12,a1,-j-p1 )  and  (tz12,all-\-pi2t>).  There  are,  therefore,  on  the 
line  2n(2n-l)  points  of  PG(2,22n)  (referred  to  as  "imaginary"'  points)  arranged 
in  2n_1  (2n-l)  pairs  of  conjugate  points  which  figure  as  the  double  points  of  the 
elliptic  transformations.     Hence  in  PG(2,22)  there  are  six  such  pairs  on  each  line. 

Ar.  in  ordinary  projective  geometry  it  can  be  proved  that  any  three  points  on  the 
line  in  PG(2,2n)  can  be  transformed  into  any  other  three  points  of  the  line  and 
that  if  three  points  of  the  line  are  fixed  all  points  of  the  lin**  are  fixed.  From  this 
it  follows  that  the  number  of  parabolic  transformations  is  22n-  1,  of  hyperbolic 
transformations  is  2"-1(2n-2)  (2n-f  1)  and  of  elliptic  transformations  is  22n(2n-l)/2. 
The  total  number  of  collineations  on  the  line  is  the  total  number  of  distinct  trans- 
formations T  having  coefficients  in  the  GF(2n)  and  determinant  not  zero.  This 
Is  determined  to  be  2n(22n-l). 

In  PG(2,22)  according  to  the  above  there  ore  60  transformations  on  the  line 
and  of  these  15  are  parabolic,  20  hyperbolic  and  24  elliptic.  Since  the  total 
group  is  of  order  60  and  can  be  exhibited  as  permutations  of  the  five  points  df 
the  line,  it  must  be  the  alternating  group  on  five  symbols  and  hence  its  subgroups 
ere  well  known.  They  will,  however,  be  enumerated  later  in  determining  the 
groups  which  leave  a  line  invariant. 

Transformations  in  the  plane.  The  general  linear  homogenous  transformation 
in  PG(2,2n)  may  be  written 

J,  :Pxi'=ailxl-\-  ai2x2+ai3x3,  i=i,2,3, 


8  U.  G.  Mitchell:  Geometry  and 

where  the  determinant  A=|«u|  of  the  transformation  is  not  zero  and  the 
domain  for  the  coefficients  is  the  GF(2n).  To  determine  the  invariant 
points   we   set   x\=xu    (i=l,2,3)    and    obtain    the   characteristic    cubic 

p>  +    («n-M„+«3s)   p2   +    (^u+^m+^ss)    p+A=o  (2) 

where  An  is  the  co-factor  of  a^  in  A—|tfjj|.  Since  there  are  2n  marks  in  the 
GF(2»)  there  are  22n(2n-l)  equations  of  the  form  (2)  belonging  to  the 
GF(2n).  Of  these  2"-l  have  all  three  roots  coincident,  (2n-l)  (2n-2)  two  roots 
coincident  and  the  other  distinct,  (2n— 1)  (2n— 2(2n— 3)/3  \  all  three  roots  dis- 
tinct, and  2"(2n-l)7'2  one  root  in  the  GF(2n)  and  two  roots  in  the  GF(22n)  but 
conjugate  with  respect  to  the  GF(2n).  These  last  are  made  up  of  the  products 
of  the  2n— 1  linear  factors  in  GF(2*)  with  the  2"(2"-1)/2  irreducible  quadratics 
which  appeared  as  characterictic  equations  of  transformations  on  the  line.  There 
are  then  2"  (2»— 1)(2B+1— l)/3cubics  (2)  having  roots  in  GF(2")  or  GF(2-»). 
The  remaining  2n(2-"-l)/3  are  irreducible  in  the  GF(2")  but  from  the  theory  of 
Galois  fields*  it  follows  that  their  roots  are  marks  of  the  GF(23n)  conjugate  with 
respect  to  the  GF(2n).    If,  therefore,  A  be  a  root  of  an  irreducible  cubic  belonging 

to  the  GF(2n)  its  other  roots  are  A2"  and  A2"",  where  A  is  a  mark  of  the  GF(23n). 
If  we  put  Xi'=xu  ( ;=1,2,3)  in  T,  and  substitute  A  for  p  we  obtain 

Xl  :^,:at3=(^]1+<72:,A+^,,A+A'')  :(412+a21k)  :(//1,,+«;1A) 
as  the  corresponding  invariant  point.  The  other  invariant  points  are  then  neces- 
sarily the  points  obtained  by  substituting  for  A  in  this  expression  A"  and  A:;  re- 
spectively. Hence  every  transformation  T,  whose  characteristic  equation  (2)  is 
irreducible  in  the  GF  (2n)  leaves  invariant  a  triangle  in  PG(2,23n)  which  will  be 
designated  as  an  imaginary  triangle  to  indicate  that  it  is  not  in  PG(2,2n). 

Corresponding  to  the  three  cases  in  which  the  characteristic  equation  (2)  has 
three  distinct  roots  there  are  then  three  types  of  transformations  having  for  in- 
variant figure  a  triangle.  These  will  be  designated  as  type  I,„  type  I,,  type  I3  ac- 
cording as  the  invariant  triangle  has  none,  one,  or  three  of  its  vertices  in  PG(2,2n). 

If  the  equation  (2)  has  two  roots  coincident  the  corresponding  collineation  is 
designated  type  II  and  leaves  invariant  two  points  and  by  duality  two  lines.  Its 
invariant  figure  has  two  points  on  one  line  and  two  lines  on  one  point.  If  the 
three  roots  of  (2)  coincide  the  corresponding  transformation  leaves  invariant  a 
lineal  element  and  is  called  type  III.  Two  special  cases  of  the«e  will  be  classified 
as  separate  types  because  of  their  importance.  A  transformation  other  than  the 
identity  which  leave*  invariant  all  points  of  a  line  /  called  the  axis  and  all  lines 
through  a  point  P  called  the  center,  is  called  a  homology  or  type  IV.  Such  trans- 
formations appear  among  the  powers  of  those. of  types  Ij  and  II.  A  transforma- 
tion, other  than  identity,  which  leaves  invariant  all  points  of  a  line  /  and  all  lines 
through  a  point  P  on  /  is  called  an  elation  or  type  V.     The  point  P  and  the  line 


Cf.  Dickson,  1.  c,  p    21  and  p.  53. 


Collin eation  Groups  of  PG(2,22).  9 

/  are  called  the  center  and  axis,  respectively,  of  the  elation.    Such  transformations 
appear  among  the  powers  of  those  of  types  II  and  III. 

The  invariant  figures  of  the  different  types  are  shown  in  Fig.  2.  Fixed  lines 
which  are  imaginary  are  dotted  and  fixed  points  which  are  imaginary  are  left 
open. 


\/ 


/    \ 


m 


The  following  formulae*   for   the  number  of  transformations  of  each  type  in 
PG(2,2n)  are  readily  obtained: 


Cf.  Dickson,  1.  c.,  pp.  237-239. 


10  U.  G.  Mitchell:  Geometry  and 

Nl0=2<n(  22"— i)2/3  • 

Nl1=2*"(23n— 1)    (2n— 1)   72 

Ni3=23n(22n-f-2n+l)    (2"+l)    (2n— 2)    (2n— 3) /« 

Nn=22n(23n— 1)    (2M-1)    (2n— 2) 

Nni=2n(23n— 1)    (22"— 1) 

Niv=22n(22n-|-2n-f-l)    (2n — 2) 

Nv=(23n— 1)    (2n-j-l) 

Identlty=l 

Total=23n  ( 23n— 1 )    ( 22n— 1 ) 
According  to  these  formulae  the  order  of  the  total  group  of  PG(2,2:)   is  60480 
distributed  as  follows: 

Ni0=19200 

Ni1=24192 

Ni3=  2240 

Nn=lC080 

Nin=3780 

Niv=    672 

Nv=     315 

Identity=  1 
Total=60480 
The  group  of  all  projective  transformations  in   PG(2,22)    will   be   designated 

aS    VJ60480' 

§4.  Cyclic  Groups  in  PG(2,22). 

We  wish  to  determine  in  detail  the  path-curves  and  periodicity  of  each  of  the 
types  in  PG(2,22).  In  so  doing  we  shall  at  the  fame  time  determine  all  of  the 
cyclic  subgroups. 

Type  I0    Consider  the  collineation 

T0:      px2'=x,-\-x2-\-x3 

The  determinant  of  T0  is  A=r.  Its  characteristic  equation  p3-\-ip2-\-i2p-\- 
r=0  is  irreducible  in  the  GF(22)  and  has  for  roots  ^47^r'9,^62  where  v  is  a 
primitive  root  of  the  GF(28)*  and  hence  v2i=i,  v63=l.  The  invariant  points 
cf  T0  are,  therefore,  A„=(l,w9,v7),  B0^-:(1,^V28)  and  C(^(l,v1V48).  T0  is  of 
period  21  and  permutes  the  points  or  PG(2,22)  in  the  order  of  their  numbering 
in  the  table  of  alignment.  There  is,  accordingly,  some  power  of  T0  which  will 
transform  any  given  point  of  PG(2,22)  into  any  other  given  point  of  PG(2,22). 
To  show  that  every  collineation  of  type   I0  in   PG(2,22)    is  conjugate   to  some 

*  For  Galois    field   tables,    see   an   article  by  W.  H.  Bussey,  in  the  Bull.  Amer.  Math.  Soc, 
Vol.  XI,  p.  27. 


Collin eatjon  Groups  of  PG(2,22).  11 

power  of  T,  it  isonly  necessary  to  show  that  any  triangle  A=  ( A,  ,A2,A3 ) ,  B=  ( A,4,A24,A84 ) , 
CblVVi  Ve)  where  X^A,  are  any  three  marks  in  the  GF(2«)  linearly 
independent  with  respect  to  the  GF(22),  can  be  transformed  into  A0,B0,C0,  re- 
spectively, by  a  transformation  in  PG(2,2n).  The  condition  that  A,,A2,A3  be  lin- 
early independent  with  respect  to  the  GF(22)  is  necessary  because  it  will  be  ob- 
served that  if  the  coordinates  of  the  point  A  satisfied  the  equation  a1xl  +a2x2-\- 
n^c^'O  those  of  B  and  C  would  also  satisfy  the  equation 

2  4 

yaxxx-\-a2x2-\-azxz)2  as  (a^^a^-^a^)2  ■«(«1;r1-f<i,*1+<i,*,)aB0, 

and  any  transformation  leaving  A,  B  and  C  invariant  would  leave  invariant  a 
line  of  PG(2,22)  and  therefore  not  be  of  type  I0. 

;— 3 

The  conditions  that  T=p*,'=2/*lj*j,     (i— 12,3),  |<7,,|   not  o,  where  every  ati 

j—l 

is  in  the  GF(22)  shall  transform  A  into  A0  are 

«.»,  A,  + « .•,oA2+fl33A3=*>,u  ( fl21A,  +tf22A2-|-fl23A3  )  )  " 
which  raised  to  the  22  and  24  powers,  are  seen  to  be  the  conditions  that  T  trans- 
form B  to  B0  and  C  and  C0.  In  (4)  we  may  assign  a3i,  a32  and  033  arbitrarily 
in  the  GF(22)  provided  not  all  are  taken  as  zero.  We  then  have  fl31Aj-(-fl32A2+ 
a33^3==vk  some  mark  other  than  zero  of  the  GF(26).  Since  A,,A2,A3  are  three 
marks  of  the  GF(26)  linearly  independent  with  respect  to  the  GF(22)  it  is  pos- 
sible to  choose  an,il2  and  alZ  within  the  GF(22)  such  that  «,,A1+fl12A2-|-«i3A3 
is  any  mark  of  the  GF(2<1.)*  Accordingly  we  take  «,,,«, 2  and  a13  such  that 
anK~\~  a\ 2^2~f-tfi3A3 =*>k"7  «'nd  similarly  «._,,,  a22  and  a2.A  such  that  tfnA1-{-AtaAs-,ras«^a 
=rk~61.  The  desired  transformation  T  is  thereby  determined  within  the 
GF(22).     Moreover  the  determinant  of  T  is  not  zero  since 

tf1IA1-r-c12A2+tf]3A3=i;k-7 
tf21A,+a22A,  -4-  fl2SA3=»k-". 
«si  A,  +a.i2\2+a3.A\3=v* 
form  a  set  of  simultaneous  non-homogeneous  equations  in  APA2  and  A3. 

The  21  powers  of  T0  form  a  cyclic  subgroup  of  G0048t  and  since  the  triangle 
ABC  can  be  chosen  in  28(2*-l)  (2*-l)/3  different  ways  there  are  960  such  conju- 
gate cyclic  subgroups  in  G60480. 

Since  the  determinant  of  T0  is  r  the  determinant  of  T02  is  i*=i  and  the  de- 
terminant of  T,,3  is  1.  The  powers  of  T0,  then,  which  are  also  powers  of 
T03  and  no  others  are  of  determinant  unity.  The  group  G21(cyc.  I„)  consisting  of 
the  21  powers  of  T0  contains  accordingly  a  self-conjugate  subgroup  of  order  7 
consisting  of  the  7  powers  of  T„3.     G„„4S1)  must  contain  960  such  cyclic  subgroups. 

Again,  T07  is  of  period  3.     Hence,  G2!    (eye.  I0)  contains  a  cyclic  subgroup  of 
order  3  which  must  also  be  self-cou jugate  since  no  others  powers  of  T„  than  pow- 
ers of  T07  are  of  period  3.    G60480  must  contain  960  such  conjugate  subgroups. 
*  Cf.  Dickson,  1.  c,  p.  49 


12  U.  G.  Mitchell:  Geometry  and 

T07  must  permute  all  points  of  PG(2,22)  in  triangles  since  if  it  permuted  any 
three  collinear  points  among  themselves  it  would  leave  invariant  the  line  joining 
them.  It  will  be  seen  later  (in  discussing  the  simple  group  G1C8)  that  a  trans- 
formation of  type  I0  of  period  7  permutes  among  themselves  seven  points  so  re- 
lated that  for  every  four  of  the  points  which  are  no  three  collinear  the  other  three 
are  the  diagonal  points  of  their  complete  quadrangle. 

Type  lv     The  collineation 

T, :      Px2'=x3 

has  the  characteristic  equation  (p-f-1)  (p2+p+l)=0  whose  roots  are  1,  u  and 
u*  where  a  is  a  primitive  root  in  the  GF(24)  and  hence  «5=j  and  «15=7.  The 
invariant  points  of  T, ,  are  Ajss  (7,0,0),  BA=  (o^w^Q^ee  (o,/,w4),  and  Tx  is 
therefore  of  type  Ix  with  A±  for  center  (or  invariant  real  point)  and  x1=0  for 
axis  ,or  invariant  real  line).  Tx  is  of  period  15  and  Tj5  and  T^0  are  homolo- 
gies. Accordingly,  the  group  G15  (eye.  IJ  consisting  of  the  15  powers  of  T1 
contains  a  self -con  jugate  cyclic  subgroup  of  order  3  containing  two  homologies 
and  the  identity.  Since  the  determinant  of  T1  is  i,  T13,T16,T19,T112,  Tt15  and  no 
other  powers  have  determinant  unity  and  are  of  period  5  with  the  exception  of 
TV^bsI.  G15(cyc.  Ix)  therefore  contains  a  self-conjugate  cyclic  subgroup  of 
order  5  consisting  of  these  transformations. 

Ti3,  which  is  of  period  5,  permutes  the  lines  through  Ax  in  cyclic  order  and 
hence  a  point  Px  not  on  the  axis  has  4  other  conjugates  P2,P3,P4,Pii  such  that  no 
two  of  the  points  P1,Po,P3,P4,P3  are  collinear  with  A,.  Moreover,  no  three  of 
the  points  P1,P2,P3,P4,P5,  can  be  collinear,  for  if  they  were  the  line  containing 
them  would  be  invariant  under  Tx8.  Hence,  P1,P2,P3,P4,Pr>  form  a  point  conic 
having  Ax  for  outside  point.  Evidently  Tj8  leaves  invariant  three  such  point 
conies  having  Ax  for  outside  point  and  by  duality  three  line  conies  having  xl^=o 
for  outside  line. 

Every  collineation  T/  of  type  l1  in  PG(2,22)  is  conjugate  to  Tx  or  some 
one  of  its  powers  since  there  is  in  PG(2,22)  a  transformation  S  transforming  any 
point  P'  and  line  /'  into  (7,0,0)  and  x1=o  respectively  and  a  transformation  Su 
leaving  the  point  (7,0,0)  fixed  and  changing  any  pair  of  conjugate  imaginary 
points  on  xl=o  into  the  pair  (o,7,m4).  The  collineation  (SS1)T/(SS1  )_1  must 
then  be  some  power  of  Tx. 

In  discussing  the  one-dimensional  transformations  it  was  shown  that  in 
PG(2,22)  there  are  six  pairs  of  conjugate  imaginary  points  on  each  line.  Hence 
there  are  in  PG(2,22)  21-16-6  =2016  conjugate  groups  G^fcyc.I,)  each  con- 
taining a  cyclic  self -con  jugate  subgroup  of  order  5  consisting  of  the  transformations 
of  period  5,  and  a  cyclic  self-conjugate  subgroup  of  order  3  consisting  of  the  hom- 
ologies. There  are  2016  of  the  cyclic  subgroups  of  order  5  but  only  21  -16=336  of 
the  subgroups  of  order  3  since  the  same  subgroup  of  order  3  appears  with  every 
G]5(cyc.I1)  which  leaves  invariant  a  given  center  and  axis. 


COLLINEATION   GROUPS  OF    PG(2,22).  13 

Type  I3.  If  T3  be  a  transformation  of  type  I,  it  leaves  invariant  a  real  trian- 
gle A,B,C  T3  is  fully  determined  by  its  invariant  triangle  and  the  transforma- 
tion of  a  point  P  into  a  point  P'  provided  the  points  A,B,C,P  andP'  are  no  three 
collinear.  Hence,  (Theorem  4,  Cor.  4)  there  are  two  choices  for  P'  for  a 
given  point  P.  Accordingly  T3  is  of  period  three  and  permutes  the  9  points  not 
on  the  sides  of  its  invariant  triangle  in  three  triangles.  It  should  be  noted  that 
any  one  of  these  triangles  together  with  the  points  A,B,C  form  a  set  of  six  points 
no  three  of  which  are  collinear  and  therefore  constitute  in  six  different  ways  a 
point  conic  and  its  outside  point.  Also  that  any  one  of  these  triangles  and  two  of 
the  points  A,B,C  form  a  point  conic  left  invariant  by  T3  and  hence  that  T,  leaves 
invariant  9  different  non-degenerate  point  conies.  If  A,B,C  be  taken  as  the  tri- 
angle of  reference  the  two  transformations  of  type  I3  which  leave  it  invariant  are 

T, :     px2=i"x2     and   T32:     px2=ix2 

Since  any  triangle  can  be  transformed  into  any  other  triangle  by  a  collineation 
within  the  PG(2,22)  it  follows  that  every  collineation  of  type  I,  is  conjugate  to 
T3  or  T32.  Since  21  -20  -16/3 1— 1120  different  triangles  can  be  chosen  in  PG 
(2,22)  there  are  1120  conjugate  cyclic  groups  G3  (eye.  I3). 

Type  II.  A  collineation  T2  of  type  II  leaves  invariant  two  real  points  A,B, 
and  a  real  line  /  (distinct  from  AB)  through  one  of  the  points,  say  A.  Two 
lines  fixed  through  A  make  the  transformation  of  lines  through  A  of  period  three. 
One  line  fixed  through  B  makes  the  transformation  of  lines  through  B  of  period 
two. 

T2  is  therefore  of  period  6,  but  only  T2  and  T25=T2  '  are  of  type  II.  T22 
and  T24  are  homologies  with  B  for  center  ?nd  /  for  axis  and  T23  is  an  elation 
with  A  for  center  and  AB  for  axis. 

If  we  select  A  as  the  point  ( 0,0,0  B  as  the  point  ( i,o,o)  and  the  line  /  as  the 
line  xi=o  we  find  that  any  point  P  not  on  AB  or  /  can  be  transformed  into  any 
other  point  P'  not  on  AB,  I.  PA  or  BB  Taking  P  and  P'  as  (/,/,/)  and  (/',/, o) 
T2  is  determined  as 

T2 :     nx2'=x2 

px/=x2-4-x:i 

On  the  line  xx=o  T2  interchanges  the  points  (o,/,o)  and  (o,i,i).  It  is  easily 
seen  that  T„  or  T.r1  is  conjugate  to  anv  other  transformation  T./  of  type  II 
in  PG(2,22)." 

Since  the  invariant  figure  can  be  chosen  in  21  •10-8=1080  different  ways 
and  for  a  given  point  P,  on  /  there  are  three  choices  for  P,'  it  follows  that 
^oo48o  contains  5040  cyclic  groups  G6  (cyc.II)  each  containing  a  self -con  jugate 
cyclic  subgroup  of  order  two  consisting  of  T,3  (an  elation)  and  the  identity,  and 
a  self -con  jugate  cyclic  subgroup  of  order  three  consisting  of  T22  and  IV  (hom- 
ologies) and  the  identity.  It  is  to  be  noted,  however,  that  each  subgroup  of 
order  two  is  common  to  16  (since  there  are  4  choices  for  B  on  AB  and  4  choices 


14  U.  G.  Mitchell:  Geometry  and 

for  /  through  A)  different  groups  G„  (eye.  II)  and  hence  that  there  are  but  315 
such  subgroups.  Also  that  each  subgroup  of  order  three  is  common  to  15  different 
groups  G6  (eye.  II)  (since  there  are  5  choices  for  A  on  /  and  3  choices  for  the  pairing 
of  lines  through  B  in  each  case)  and  hence  there  are  but  336  different  such  sub- 
groups of  order  three. 

Type  III.  A  transformation  T  of  type  III  leaves  invariant  a  line  /  and  a 
point  A  on  /.  Since  one  line  through  A  is  fixed  the  transformation  of  lines  through 
A  is  parabolic.  T2  is  therefore  an  elation  and  T*  must  be  the  identical  trans- 
formation. T  is  consequently  of  period  4  and  permutes  four  points,  no  one  on  / 
and  no  three  of  which  are  collinear,  in  cyclic  order.  Since  the  transformation 
of  four  points  no  three  of  which  are  collinear  into  four  such  points  fully  de- 
termines a  projective  transformation  it  follows  that  a  transformation  T  of  type 
ill  is  fully  determined  by  any  four  such  points  which  T  permutes  in  cyclic  order. 

The  collineation  of  type  III  determined  by  permuting  the  four  points   (7,0,0), 

( I>I>o)>(i>0,i)f.(i,i,i),  no  three  of  which  are  collinear,  in  cyclic  order  as  named  is 

f»*i'— *i 
T:    px2'=x1-\-x2 

Px./=x2-\-xz 
T  leaves  invariant  the  point  (0,0,1)  and  the  line  x1=0.      It  is  readily  seen  that  in 
PG(2,22)  every  collineation  of  type  III  is  conjugate  to  either  T  or  T3. 

Four  noncollinear  points  A,B,C,D  can  be  chosen  in  21  -20  -16 :9/4  !=2520  dif- 
ferent ways.  Each  cyclic  order  determines  a  transformation  of  type  III  not  a 
power  of  any  determined  by  any  other  cyclic  order  and  each  transformation  of 
type  III  permutes  in  cyclic  order  the  points  of  four  different  quadrangles.  It 
follows  therefore  that  there  are  2520-3/4=1890  cyclic  groups  G4  (eye.  Ill)  in 
^60480  eacn  containing  a  self -con  jugate  cyclic  subgroup  of  order  two.  It  is  to  be 
noted  that  a  subgroup  of  order  two  is  common  to  6  different  groups  G4(cyc.  Ill) 
and  hence  that  there  are  but  315  such  groups. 

Type  IF.  Homologies.  The  homologies  in  PG(2,22)  have  appeared  as  the 
336  cyclic  subgroups  of  the  2016  G15  (eye.  IJ  and  the  5040  G6  (eye.  II).  It 
was  shown  that  the  336  cyclic  G3  (eye.  IV)  were  conjugate  under  the  group 
G6048o-  A  homology,  as  has  been  seen,  is  of  period  three  and  its  path-curves  are  the 
straight  lines  through  the  center.     The  homology 

,;xl/=ix1 
T :     fix2'=x2 

may  be  taken  as  a  canonical  form. 

Type  V.  Elations.  The  elations  have  appeared  as  315  conjugate  cyclic  sub- 
groups of  order  two  in  both  the  5040  groups  G6  (eye.  II)  and  the  1890  groups 
G4  (eye.  III).  Each  elation  is  of  period  2  and  its  path-curves  are  the  straight 
lines  through  the  center.     The  elation 

fJx/=x, 
T:     px2=x2 

px./=x}-{-x3 


Collin eation  Groups  of  PG(2,2-).  15 

which  has  for  center  the  point  {0,0,1)  and  for  axis  the  line  xl==o  may  be  taken 
as  a  canonical  form. 

§5.     The  Group  of  Determinant   Unity  — G20160. 

Theorem  5.  In  PG  (2,2")  every  group  G  of  order  N  which  contains  col- 
lineations of  determinant  not  unity  contains  exactly  N/3  collineations  of  determi- 
nant unity. 

Proof.  It  is  obvious  that  the  collineations  of  determinant  unity  in  G  form  a 
self-conjugate  subgroup  Gn.  Suppose  n  greater  than  N/3  .  If  T  be  any  collinea- 
tion  in  G  but  not  in  Gn  the  products  of  T  and  TJ  by  the  n  collineations  in 
Gn  are  2n  distinct  collineations  and  G  would  contain  3n>N  distinct  collinea- 
tions which  is  contrary  to  hypothesis.  Suppose  n  to  be  less  than  N/3 .  G  must 
then  contain  m>  N/3  collineations  of  determinant  d  where  d  is  either  i  or  r. 
If  T  be  any  collineation  in  G  of  determinant  d2  the  products  of  the  m 
collineations  of  determinant  d  by  T  are  m>N/3  distinct  collineations  of  determi 
riant  unity  in  G,  contrary  to  supposition.  Since  n  is  neither  less  nor  greater  than 
N/3  it  follows  that  n=  N/3 

The  Group  of  Determinant  Unity.  By  Theorem  5  the  group  G60480  has  a  self-con- 
jugate subgroup  of  determinant  unity  of  order  60480/3=20160.  In  §4  it  was  shown 
that  all  collineations  of  types  Ia,III,  V  and  those  of  type  I0  of  period  7  and  type 
I  of  period  5  were  of  determinant  unity.  Hence  the  Group20160  of  determinant 
unity  contains  the  following  collineations: 

The    identical    collineation 1 

All   collineations  of   type   I3 2240 

Those  collineations  of  type  Ii  which  are  of  period  5, 

(1-3  of  the  total  number) 8064 

Those  collineations  of  type  I0  which  are  of  period  7, 

(3-10  of  the  total  number) 5760 

All  collineations  of   type   III 3780 

All   collineations  of   type  V 315 


20160 
The  group  will  be  designated  as  above  by  G20180.     It  has  been  proved  that  in  any 
PG(k,pn)    the   group   of   all   collineations   of   determinant    unity   is   the   maximal 
simple  subgroup  of  all  collineations  in  the  PG(k,pn).* 

Theorem  6.      Every    collineation  in    G2inM    can    be    obtained    as    a    product* 
of  elations. 

Proof.     Let  T    be    any    collineation    of    type  I3    determined    by  the    equation 


*  Cf.  VEBLENand  Bussey,  1.  c,  p.  253  and  Dickson,  1.  c,  p.  87. 


16  U.  G.  Mitchell:  Geometry  and 

T(A1A2A3A4)=A1A2A3A5  where  no  three  of  the  points  A1,A2,A3,A4,A5  are 
collinear.     Two  elations  E1  and  E2  are  determined  by  the  following  equations: 

Ei(AiA2A3A4)  =  AtA3A2A4 

E2(A2A3A4A5)=A0A2A5A4 
such  that  their  product  E2E1=T.  That  Et  and  E2  are  elations  follows  from  the 
facts  that  elations  are  the  only  collineations  in  PG(2,22)  of  period  two  and  that 
the  points  A1,A2,A3,A4,A3  are  no  three  collinear.  Since  the  five  points  are  no 
three  collinear  they  form  a  conic  and  since  E2  interchanges  four  of  the  points  by 
pairs  it  leaves  invariant  point  by  point  the  diagonal  line  of  the  complete  quadran- 
gle of  the  four  points.  By  Corollary  5  of  Theorem  4  this  line  contains  the  fifth 
point  of  the  conic.  E2,  therefore,  leaves  Ai  invariant  and  it  is  clear  that  E2EL 
-=  T.  Hence  every  collineation  of  type  I3  can  be  obtained  as  the  product  of  two 
elations. 

Let  Tj  be  any  collineation  of  type  I,  of  period  5  determined  by  the  equation 
Tx  (A1A.,A3A4)  =  A2A3A4A5, 
where  A1,A2,A3,A4,A5    are   five    points   no   three   of   which   are   collinear.      Two 
elations  E/  and  E2'  are  determined  by  the  equations 

E1'(A1A,A4A,)=A5A4A2A1) 
E/  ( A2AaA4A5 )  =  A5  A4A3A2,     ' 
such  that  E2,E/=T].     That  E/  is  an  elation  leaving  A3  invariant  and  that  E2 
is  an  elation  leaving  A1  invariant  follows  by  the  reasoning  given  above  to  show 
that  E2  was  an  elation  leaving     A1  invariant.        Hence  every  collineation  Tl  of 
type  Ix  of  period  five  can  be  obtained  as  the  product  of  two  elations. 
The  transformation 

T0:     px/=Ari-|-*, 
is  of  type  I0  of  period  7.     It  is  found  that  T0=  TXE  where  E  is  an  elation, 

E:      ox./=x., 

and  Tx  is  of  type  I2  of  period  five 

px/-=x1-\-x., 
T, :     ,'.v./=x3 

pX./=X.,-\-X:i 

But  since  every  collineation  of  type  Ix  of  period  five  can  be  obtained  as  a  product 
of  elations  and  T0  or  one  of  its  powers  is  conjugate  to  every  collineation  of  type 
I0  of  period  seven  within  the  PG(2,22)  it  follows  that  every  collineation  of  type 
I0  of  period  seven  can  be  obtained  as  a  product  of  elations. 

Let  S  be  any  collineation  of  type  III  determined  by  the  equation 

S  ( Ax  A2  A3  A4 )  =  A2  A3  A4  Aj , 

where  no  three  of  the  points  A1,A2,A3,A4  are  collinear.  Any  transformation  so 
determined  must  be  of  type  III  because  in  PG(2,22)   transformations  of  type  III 


COLLINEATION    GROUPS  OF    PG(2,22).  17 

and  no  others  are  of  period  four.     Then  S=E2E1  where  Et  and  E,  are  elations 
determined  by  the  equations 

E1(A1A2A;,A4)  =  AjA4A3A2> 

K2(A1A2A3Ai)  =  A2A1A4A3. 

E!  and  E2  are  again  necessarily  elations  because  elations  are  of  period  two  and 
the  points  are  no  three  collinear. 

Since  every  collineation  not  an  elation  in  G20i60  must  be  of  type  I3,  Ij  (of  pe- 
riod 5),  I0  (of  period  7),  or  III  the  theorem  is  established. 

Theorem  7.     In  PG(2,22)   if  a    group    Ga    of    determinant    unity    be  trans- 
itive on  all  points  and  lines  of  the  plane  and  contain  a  single  elation  it  contains 
all  elations. 

Proof.  In  PG(2,22)  three  and  but  three  elations  have  the  same  center  and 
axis  since  there  are  but  three  ways  in  which  the  four  points  other  than  the  center 
on  an  invariant  line  can  be  paired.  The  theorem  will  follow,  therefore,  if  it  can 
be  shown  that  if  Ga  contain  a  single  elation  it  must  contain  elations  such  that  for 
any  given  line  /  and  point  P  on  /  there  are  three  elations  in  Ga  having  P  for 
center  and  /  for  axis. 

From  the  transitivity  of  Ga  it  follows  that  Ga  must  contain  transforms  of  the 
given  elation  such  that  every  point  in  the  plane  is  the  center  and  every  line  the 
axis  of  at  least  one  elation.  Also  the  order  of  Ga  must  be  a  multiple  of  21  and 
therefore  Ga  must  contain  a  collineation  of  period  3.  Since  the  only  collineations 
in  PG(2,2')  of  determinant  unity  and  period  three  are  of  type  I3  it  follows  that 
Ga  must  contain  collineations  of  type  I3  such  that  every  point  in  the  plane  is  a 
vertex  and  every  line  of  the  plane  is  a  side  of  the  invariant  triangle  of  at  least 
one  collineation  of  type  I3. 

For  the  given  line  /,  then,  there  is  in  Ga  an  elation  E  having  /  for  axis  and  a 
collineation  T  of  type  I3  having  /  for  an  invariant  line.    Four  cases  may  arise, 
(a).     P  may  be  the  center  of  F  and  an  invariant  point  of  T. 

Since  T  leaves  invariant  a  point  which  E  transforms  they  cannot  be  commuta- 
tive and   hence  TET-1    and  T-ET2  are  the  other  two  elations  having  P  for  cen- 
ter and  /  for  axis, 
(b).     P  may  be  the  center  of  E  and  not  an  invariant  point  of  T. 

Let  A  and  B  be  the  invariant  points  on  /  of  T.  A  must  be  the  center  of  some 
elation  Ej.  If  /  be  not  the  axis  of  Et  we  have  E1TE1_1=T1  a  collineation  of 
tjpe  I3  having  A  and  some  point  B'  different  from  B  on  /  for  invariant  points.  By- 
transforming  T,  through  the  power  of  T  which  transforms  B'to  P  the  case  is  reduced 
to  case  (a).  A  similar  argument  applies  to  the  point  B.  If  neither  A  nor  B  be 
the  center  of  an  elation  whose  axis  is  not  /,  by  case  (a)  Ga  must  contain  all 
elations  having  A  or  B  for  center  and  /  for  axis.  The  three  elations  E,,  E2,  Es 
having  A  for  center  and  /  for  axis  form  with  the  identity  a  group  since  the 
product  of  any  two  of  them  is  an  elation  having  /  for  axis  and  A  for  center. 
Similarly  the  three  elations  E,',F./,E.,  having  B  for  center  and  /  for  axis  form 


18  U.  G.  Mitchell:  Geometry  and 

a  group.  The  nine  products  EjE/  are  all  distinct  since  if  EiEj,=EkE1/  it  fol- 
lows that  Ej'E/— EjEk  which  cannot  be  true.  Moreover,  every  EiE/  is  an 
elation  having  /  foi  axis  since  it  is  of  determinant  unity,  leaves  fixed  every  point 
of  /  and  can  not  be  the  identity.  Since  the  nine  elations  Es  E/  are  all  distinct  and 
have  /  for  axis  they  include  the  three  elations  having  P  for  center  and  /  for  axis, 
(c).  P  may  not  be  the  center  of  E  and  may  be  an  invariant  point  of  T. 
P  must  then  be  the  center  of  some  elation  E'  having  some  other  line  than  /  for 
axis.  The  transforms  of  E  through  E',  T  and  T2  give  elations  such  that  every 
point  of  /  other  than  P  is  the  center  of  an  elation  having  /  for  axis.  Since  the  lines 
through  P  can  be  interchanged  by  pairs  in  three  ways  only,  the  product  of  some 
two  of  these  four  elations  is  an  elation  with  P  for  center  and  /  for  axis.  This  case 
is  thereby  reduced  to  case  (a). 

(d).  P  may  be  neither  the  center  of  E  nor  an  invariant  point  of  T.  If  C,  the 
center  of  E,  be  not  an  invariant  point  of  T  by  transforming  E  through  T  or  T2 
•'whichever  transforms  C  to  P)  the  case  is  reduced  to  case  (b).  If  C,  the  center  of 
E,  be  one  of  the  invariants  points  of  T  we  may  transform  E  through  Elt  the  elation 
having  P  for  center  and  some  other  line  than  /  for  axis,  and  obtain  an  elation  E2 
having  some  other  point  on  /  for  center.  If  E2  have  for  center  the  other  invariant 
point  of  T  the  product  EE2  is  an  elation  E3  whose  center  is  not  one  of  the  invariant 
points  of  T.  The  transforming  of  E2  or  E3  through  T  or  T2  then  reduces  this 
case  as  above  to  case  (b),  and  completes  the  proof  of  the  theorem. 

Definition  of  Figure.  In  PG(2,2n)  a  point  figure  is  defined  as  any  set  of  m 
points  where  m  is  any  positive  integer  less  than  22n— |-2n— (-1.  Similarly  a  line  figure 
consists  of  any  m  lines.  The  term  figure  is  used  to  refer  to  either  a  point  figure  or 
a  line  figure.  A  real  figure  in  PG(2,2n)  is  a  figure  all  of  whose  points  and  lines 
belong  to  the  PG(2,2"). 

It  is  obvious  that  any  subgroup  of  G00480  which  leaves  invariant  no  real  figure  is 
transitive  on  all  points  and  lines  of  the  plane. 

Theorem  8.  There  is  no  subgroup  of  G20160  which  does  not  leave  invariant  a 
real  figure  on  an  imaginary  triangle. 

Proof.  Any  such  subgroup  Gk  can  contain  no  elation,  for  by  Theorem  7  if  Gk 
contained  a  single  elation  it  would  contain  all  elations  and  hence,  by  Theorem  6,  all 
collineations  in  G20]60.  Also  Gk  can  contain  no  collineation  of  type  III  since  the 
square  of  a  type  III  is  an  elation. 

Suppose  Gk  to  contain  a  collineation  T,  of  type  lx  and  let  its 
center  be  designated  Pr  As  was  seen  in  the  proof  of  theorem  7,  since 
Gk  is  transitive  and  of  determinant  unity  it  contains  some  collineation 
T3  of  type  I3  which  leaves  Px  invariant.  Let  lx  and  l2  be  the  two  lines  through 
Px  left  invariant  by  T3.  Since  T1  is  of  period  5  on  the  lines  through  Px  some  power 
of  T1(  say   Txm    transforms  /x   to  l2.  Let  /3,/4,/5    be  the  lines  into  which  T\m 

tiansforms  /2,/3,/4  respectively.  Some  power  of  T3,  say  T3n,  produces  among  the 
lines  through  Y>1  the  transformation  (lx)  (I.,)  (IJJ^)-  Hence  the  product 
T,2mT3n  produces  among  the  lines  through  Px  the  transformation   (lj3)  (LI*)  (l5) 


COLLINEATION   GROUPS  OF    PG(2,22).  19 

The  collineation  T,2mT3n  leaves  invariant  the  point  P,  and  a  single  line  /.  through 
Pr  Such  a  collineation  must  be  of  type  III  or  an  elation.  Hence  Gk  can  contain 
no  collineation  of  type  Ir 

Since  the  only  other  collineations  of  determinant  unity  are  of  type  I3  (of  period  3) 
and  type  I0  (of  period  7)  Gk  can  contain  only  collineations  of  these  two  types. 
Since  20160— x2*'3a'5r7  and  the  order  of  Gk  must  be  divisible  by  21  the  only  possible 
orders  for  Gk  are  21  and  63.  But  as  a  consequence  of  Sylow's  Theorem*  any  group 
of  order  21  or  G3  must  contain  a  self -con  jugate  cyclic  subgroup  of  order  7  since  the 
order  of  the  group  can  be  written  in  the  form  7m(l-f-7k)  where  7m  is  the  order 
of  the  largest  group  within  which  the  cyclic  subgroup  of  order  7  is  self- 
conjugate  and  1  -j-7k  is  the  number  of  cyclic  subgroups  of  order  7.  For  order  21 
the  only  possibility  is  k=0  and  m=3  and  for  order  63  k  =0  and  m=9.  In 
PG(2,2-)  the  only  possible  cyclic  group  of  order  7  is  a  G7  which  leaves  invariant 
an  imaginary  triangle  F,.  But  if  the  G-  be  self -con  jugate  within  the  Gk  every 
collineation  in  Gk  must  leave  invariant  the  imaginary  triangle  Fi.  Hence  there 
is  no  subgroup  Gk  of  G201C0  which  does  not  leave  invariant  either  a  real  figure  or 
an  imaginary  triangle. 

Theorem  P.  There  is  no  subgroup  of  G60480  except  G20]60  which  does  not 
leave  invariant  a  real  figure  or  an  imaginary  triangle. 

Proof.  If  any  subgroup,  say  Gn,  exist  it  must  contain  a  self-conjugate  sub- 
group Hn  of  determinant  unity  which  leaves  invariant  no  real  figure  or  imaginary 
triangle  contrary  to  theorem  8. 

§6.     The  Group  Leaving  Invariant  an  Imaginary   Triangle — G6S. 

Theorem  10.  The  group  of  all  collineations  in  PG(2,22)  which  leave  inva- 
riant a  given  imaginary  triangle  Ft  is  of  order  63. 

Proof.  Let  the  group  be  designated  Ga.  In  §4  it  was  shown  that  if  a  collin- 
eation leave  fixed  one  vertex  of  Fi  it  leaves  fixed  every  vertex  of  F(.  Hence  every 
collineation  leaving  Fi  invariant  must  either  permute  the  vertices  of  F|  in  cyclic 
order  or  leave  each  vertex  fixed.  It  was  also  shown  in  §  4  that  there  are  exactly 
21  collineations — the  21  powers  cf  a  type  I()  of  period  21 — which  leave  each  vortex 
of  an  imaginary  triangle  Ft  fixed.  That  there  can  not  be  more  than  21  such  col- 
lineations follows  from  the  fact  that  ?  collineation  is  fully  determined  by  the 
leaving  fixed  of  each  vertex  of  an  imaginary  triangle  and  the  transformation  ot 
one  real  point  into  another  real  point. 

There  can  not  be  more  than  21  collineations  permuting  the  vertices  AiBiCi  of  F|  in 

a  given  cyclic  order  (AiBiC,).     For  suppose  S,,  S„  S„, Sn  to  be  n  such 

collineations  where  n>21.     Then  if  T  be  a  collineation  permuting  Ai,B1(Ci   in 

the  order  (AiCiBt)  there  are  within  G,  n">21  collineations  TSj,TS2,  TS3 

,TSn,  distinct  from  each  other  and  each  leaving  every  vertex  A|,B,,C| 


*  See  Burnside,  Theory  of  Groups,  p.  94. 


20  U.  G.  Mitchell:  Geometry  and 

fixed,  contrary  to  the  hypothesis  that  Ga  contains  but  21  collineations  leaving  each 
vertex  of  Fi  fixed. 

That  there  exists  a  group  of  order  63  leaving  Ft  invariant  is  shown  by  consid- 
eration of  the  transformations 

/»*i'=**i+*3  Pxi'=xiJrx3 

T0:     Px./=xi-\-ix2  and  T3:      ^v/=x3 

Pxz'=x2-yr-xz  p*3'=*2+*3 

T0  is  of  type  I0  of  period  21  and  T3  is  of  type  I3  of  period  3.  Tn  leaves  fixed 
each  vertex  Ai=(j,*;27,vs6),  Bi^(i,viS,v  18),Ci=  (i,vr"l,v9)  [where  v  is  a  prim- 
itive root  of  the  GF(26)]  of  Fi,  and  T3  permutes  these  vertices  in  the  order 
(A^Q).  T0  and  T3,  therefore,  generate  a  group  of  order  63  leaving  invariant 
the  imaginary  triangle  Fj.  Since  it  was  shown  in  §3  that  F(  can  be  transformed  into 
any  other  imaginary  triangle  Fi  by  a  collineation  within  the  PG(2,2'-),  there  is  a 
group  of  order  63  leaving  invariant  any  such  triangle. 

Theorem     11.     The  only  groups  in  PG(2,22)   which   leave  invariant  an  im- 
aginary triangle  Fi  are  the  following: 

A.  Groups  leaving  each  vertex  of  Fi  fixed. 

a.  A  cyclic  group  G3(cyc.I0)  of  collineations  of  type  I0  of  period  J. 

b.  A  cyclic  group  G7(cyc.  /„)   of  collineations  of  type  I0  of  period  7. 

c.  A  cyclic  group  G21(cyc.  I0)  of  collineations  of  type  I0  of  period  21. 

B.  Groups  permuting  the  vertices  of  F%. 

a.  A  cyclic  group  G3(cyc.  70)   of  collineations  of  type  I0  of  period  J. 

b.  A  cyclic  group  G3(cyc.  I0)  of  collineations  of  type  /0  of  period  J. 

c.  An  Abelian  group  G&  leaving  invariant  also  a  real  triangle,  and  contain- 
ing besides  the  identity  6  collineations  of  type  I0  of  period  J  and  2  col- 
lineations of  type  I3. 

d.  A  self-conjugate  group  G21  of  determinant  unity  containing  besides  the 
identity  6  collineations  of  type  I0  of  period  J  and  14  collineations  of  type  I3. 

e.  A  group  G6Z  of  all  collineations  in  PG(2,22)  which  leave  Fi  invariant 
containing  besides  the  identity  6  collineations  of  type  I0  of  period  7,  JO  of 

type  I0  of  period  J,  12  of  type  I0  of  period  21,  and,  14  of  type  I3. 
Proof.  The  existence  of  the  group  G„a  of  all  collineations  in  PG(2,2-)  leaving 
l'j  invariant  was  shown  in  the  proof  of  the  preceding  Theorem.  The  existence  of 
the  cyclic  subgroups  is  obvious  and  the  existence  of  the  G21  of  determinant  unity 
follows  from  Theorem  5.  The  group  Gn  is  a  Sylow  subgroup  and  that  it  is  Abe- 
lian follows  from  the  fact  that  its  order  is  the  square  of  a  prime.*  To  establish 
the  Theorem  it  is  only  necessary  to  show  further  that  every  subgroup  of  the  G63  of 
all  collineations  leaving  Fs  invariant  is  one  of  the  kinds  enumerated  above.  The 
only  possible  orders  for  such  subgroups  are  3,  7,  9,  and  21.  All  subgroups  of  order 
3  or  7  must  be  among  the  cyclic  groups  enumerated  above  since  3  and  7  are  primes. 
A  group  of  order  9  must  be  Abelian  and  by  Theorem  5  must  contain  a  G3(cyc.  I3) 


Cf.  Burnside,  1.  c,  p.  63. 


Collin eation  Groups  of  PG(2,22).  21 

of  determinant  unity.  But  no  such  G9  can  contain  more  than  one  G3(cyc.  I„)  ; 
for  if  T,  and  T2  be  two  collineations  of  type  I,  which  do  not  belong  to  the  same 
G3(cyc.  I3)  the  product  of  Tt  by  the  power  of  T2  which  permutes  the  vertices 
of  Fi  in  inverse  order  is  a  collineation  of  determinant  unity  leaving  each  vertex  of 
Fj  fixed  and  therefore  of  type  I0  of  period  7.  Hence  every  subgroup  of  G„3  of 
order  9  contains  a  self-conjugate  G8(cyc.  I3)  and  leaves  invariant  a  real  triangle. 
A  subgroup  of  G03  of  order  21  must  be  the  direct  product  of  a  G3  and  a  G7.  Since 
the  GT  must  be  a  G7  (eye.  I0)  and  the  G.,  must  be  either  a  G3(cyc.  I0)  or  a  G, 
(eye.  I3)  every  such  subgroup  must  be  either  a  G21(cyc.  I0)  or  a  G2l  of  determinant 
unity  and  therefore  one  of  the  kinds  enumerated  in  the  Theorem. 

§  7.  Invariant  Real  Figures  and  Their  Groups, 

It  has  now  been  shown  that  every  subgroup  of  the  G00480  except  the  self-conju- 
gate G2l)]00  of  determinant  unity  leaves  invariant  a  real  figure  or  an  imaginary 
triangle,  and  every  group  which  leaves  invariant  an  imaginary  triangle  has  been 
determined.  Accordingly  we  next  take  up  the  question  of  determining  what  real 
figures  can  be  the  invariant  figures  of  groups  in  PG(2,22)  and  what  group  or 
groups  leave  each  invariant.  In  determining  these  groups  it  is  sufficient  to  con- 
sider point  figures;  for,  since  a  collineation  in  the  plane  is  self-dual,  corresponding 
to  every  group  which  leaves  invariant  an  //-line  figure  there  is  a  group  of  the  same 
order  which  leaves  invariant  the  dual  //-point  figure.  Abstractly  considered  the 
two  groups  are  identical.  Furthermore,  in  PG(2,22)  it  is  sufficient  to  consider 
point  figures  in  which  the  number  of  points  n  is  less  than  11,  for  if  n  ==  11  the 
point  figure  consisting  of  21— n  (or  some  lesser  number)  can  be  taken  as  the  inva- 
riant figure  of  the  group. 

In  this  section  will  be  determined  all  groups  which  leave  invariant  real  point- 
figures  whose  points  are  not  all  collinear  and  which  leave  no  point  fixed  under 
all  transformations  of  the  group.  To  obtain  all  such  groups  it  is  only  necessary  to 
determine  for  each  value  of  n  from  n  =  10  to  n  ==  3  all  groups  which  are  trans- 
itive on  all  points  of  the  //-point  figure ;  for,  if  such  a  group  be  not  transitive  on 
all  points  of  an  //-point  invariant  figure  it  must  appear  as  a  group  which  is  trans- 
itive on  an  w-point  figure  where  3<s/w<//. 

A  group  which  leaves  invariant  an  //-point  figure  also  leaves  invariant  an  as- 
sociated line  figure  each  line  of  which  contains  the  same  number  of  points;  for, 
every  line  containing  k  of  the  //  points  can  be  transformed  by  a  collineation  within 
the  group  into  some  line  through  each  of  the  other  n-k  points  and  hence  each 
of  the  n  points  lies  on  at  least  one  liw  containing  k  of  the  n  points.  It  is,  of 
course,  obvious  that  every  transform  of  a  line  which  contains  k  of  the  n  points 
must  also  contain  k  of  the  n  points  if  the  //point  figure  is  invariant  under  the  group. 
It  follows  by  the  same  reasoning  that  through  each  of  the  n  points  there  must  be 
the  same  number  of  lines.     Hence  the  figure  made  up  of  an  //-point  and  its  asso 


22 


U.  G.  Mitchell:  Geometry  and 


dated  m    line    may    be    called    a    configuration*    and    represented    by  the    symbol 


I   n    I 
k   m 


where  n  is  to  indicate  the  total  number  of  points,  m  the  total  number 
of  lines,  k  the  number  of  paints  on  each  line  and  /  the  number  of  lines  through 
each  point  of  the  configuration.  In  such  a  configuration  the  points  and  lines  are 
so  related  that  nl=km,  and  \ -\- l{k-V)^>n.  Also,  since  in  PG(2,22)  not  more 
than  6  points  can  be  chosen  such  that  no  three  are  collinear  (Cor.  4,  Theorem  4) 
if  n  >  6,  k  <£  3.  Since  not  more  than  6  lines  can  be  chosen  such  that  no  three  are 
concurrent  it  follows  that  if  m  >6  either  /<£3  or  ra<£3£. 

By  making  use  of  the  above  relations  and  the  fact  that  whenever  m<«  or 
21 — m<n  it  follows  by  duality  that  the  same  group  must  appear  as  a  group  leaving 
invariant  a  lesser  number  of  points,  we  find  that  the  possible  configurations  in 
PG(2,22)  reduce  to  the  following: 

(a) 


(d) 


(g) 


(J) 


10 
3 

3 
10 

—  F 

y  10)3 

V 

u;    1 
W; 

(h)| 
1 

(k)j 
_F 

0 
3 

4 
12 

UF 

1           x  9'3 

j=F7„ 

— F    " 

j=F„2' 

(2 

2 
3 

^|  9 

1   3 

3 
9 

=F 

l-F 

x  8>3 

1  2 

5 

15  | 

1  8 
I3 

3 

8 

7 
3 

3 

7 

=F 

x  G>2 

6 
2 

3 
9 

1  2 

2  1 
6  1 

(  6 
!  2 

4 
12 

— F    ' 

1  6>2 

=F0,2 

=Fr„2 

5 
2 

r 

V2 

2 
5 

<1>P 

|2 

l-F 

1        3'2' 

3 
6 

1  5 
1   2 

4 
10 

K 

1    2 

2 
4 

It  is  observed  that  a  group  which  leaves  invariant  an  F^j  also  leaves  invariant 
the  figure  made  up  of  the  remaining  points  and  lines  of  the  plane.  This  figu;v 
will  be  called  the  residual  figure  and  referred  to  as  Rj„. 

We  will  consider  these  configurations  in  order. 


(a) 


3     10 
10     3 


=-F 

L  10) 


Consider 


point       P       of       F10,3       and       the       three     lines     lx,l^,lz     of 


Ci.  Veblen  and  Young,  Projective  Geometry,  Vol.  I,  pp.  38-39. 


Collin eation  Groups  of  PG(2,2-), 


23 


F„„3  which  pass  through  P.  On  lxJJ:i  are  7  points  of  F,,,,.,  and  hence  there  are 
three  points  PlfP8,P,  of  F,0M  not  on  any  of  the  lines  l^l.J^.  This  necessitates 
either  that  G  lines  lx,iJA,  PPltPPL„PP3  pass  through  P  or  that  two  or  more  of  the 
points  Pl,Pa,P,  are  collinear  with  P.  Since  neither  of  these  conclusions  is  allow- 
able under  our  hypotheses,  Fx0„  is  not  a  possible  configuration  in  PG(2,2*). 

(b) 


4 
12 


On  each  line  of  F0,3  must  be  two  points  which  do  not  belong  to  FD,3.  Let  any 
line  of  F,„  be  chosen  as  the  line  xl=o  and  the  two  points  on  it  which  do  not 
belong  to  F9,3  as  the  points  12  (o,o,j)*  and  17  (o,i,o).  Then  the  points  0 
(o,i,i),  10  (0,1,1),  and  18  (o,i,i)  on  x,=o  must  be  points  of  F0,3.  Through  0 
passes  one  and  but  one  line  which  does  not  belong  to  F0,3.  Let  the  point  of  inter- 
section of  this  line  with  the  similar  line  through  10  be  chosen  as  the  point  4 
(1,0,0).  Neither  of  these  lines  can  contain  any  other  points  of  F9,3  than  0  and 
-0,  respectively,  because  the  other  three  lines  through  either  point  contain  six  other 
points  of  F9,3  which  added  to  the  three  points  on  xx=o  gives  the  total  nine  points 
of  Fn,3.  The  point  4  is  therefore  not  a  point  in  F9,3.  Now  no  line  through  12 
which  does  not  pass  through  4  can  be  a  line  of  F9,3  since  such  a  line  contains  at 
least  three  points  (12  and  its  two  intersections  with  the  lines  from  4  to  0  and  10, 
respectively,)  not  in  F9,3.  A  similar  argument  applies  to  the  point  17.  But  every 
line  of  F9,3  passes  through  some  point  of  xx=o  and  but  nine  besides  xt=o  pass 
through  the  points  0,  10,  18.  Hence  the  lines  x2=o  and 
x2=o  are  lines  of  F9„  and  the  nine  points  of  F9,3  lie 
three  by  three  on  the  sides  of  the  triangle  of  refer- 
ence On  the  side  x2=o  are  the  points  6  (i,o,i), 
11  (1,0,1),  15  (i,o,i),  and  on  x3=o  are  3 
(1,1,0),  7  (i,i,o),  19  (1,  i,o).  A  reference 
to  the  table  of  alignment  (p.  3)  shows  that 
these  nine  points  are  collinear  by  threes 
on  nine  other  lines  as  shown  in  the 
accompanying  figure  (Fig.  3). 
Since  xl=o  Avas  chosen 
as  any  line 


Fijj ure  3 


*  Numbers  printed  thus,  12,  17,  etc.,    refer   to    the    numbers  assigned  to  points  with  certain 
coordinates,  as  given  in  the  table  of  alignment  on  p.    3 


24  U.  G.  Mitchell:  Geometry  and 

in  F0)3  it  follows  that  through  each  point  not  belonging  to  F9,3  pass  two  lines  of 
F11)3  and  three  lines  not  belonging  to  Fs„3,  anu  each  une  not  belonging  to  F0,a 
contains  one  and  but  one  point  of  F9,3. 

Having  found  that  F9,3  is  a  possible  configuration  in  PG(2,2-)  we  next  pro- 
ceed to  determine  what  collineations  can  leave  it  invariant. 

No  line  of  F9,3  can  be  the  axis  of  an  elation  leaving  F9,3  invariant,  for  an 
elation  interchanges  all  lines  not  invariant  by  pairs.  Hence  not  more  than  one 
point  of  F0,3  can  be  invariant  under  an  elation.  But  since  an  elation  interchanges 
all  points  not  invarianl  by  pairs  at  least  one  of  the  nine  points  of  F9)3  must  be 
invariant  under  an  elation  which  leaves  F„,3  invariant.  If  any  point  of  F9,3  is  to 
be  the  center  of  such  an  elation  the  axis  of  the  elation  must  be  the  one  line  through 
the  point  which  contains  no  other  point  of  F9,3,  that  is,  the  axis  must  be  the  line 
joining  the  point  to  the  opposite  vertex  of  the  triangle  of  reference.  An  elation 
having  such  a  center  and  axis  and  interchanging  the  other  two  vertices  of  the 
triangle  of  reference  must  leave  F9,3  invariant  since  the  nine  point*  of  F9,3  lie 
three  by  three  on  the  sides  of  the  triangle  of  reference.  It  is  obvious  that  there 
exists  one  and  but  one  such  elation*  for  each  point  in  F9,3.  Moreover,  no  point 
of  the  residual  figure  R9,3  can  be  the  center  of  an  elation  leaving  F9,3  invariant 
lor  through  such  a  point  pass  two  lines  of  F9,3  upon  each  of  which  are  three  points 
cf  F9,3  which  could  not  be  interchanged  by  pairs.  There  are,  therefore,  nine  and 
but  nine  elations  in  PG(2,22)  which  leave  F(l,3  invariant. 

If  T  be  a  transformation  of  type  I3  which  leaves  F9,3  invariant  no  point  P  of 
F,,,3  can  be  a  vertex  of  its  invariant  triangle;  for  at  least  one  of  the  invariant  lines 
through  P  would  have  to  be  a  line  of  F9)J  and  on  that  line  a  point  of  F9,3  would 
be  transformed  into  a  point  in  R9,3.  If  any  point  in  R0,3  can  be  a  vertex  of  the 
invariant  triangle  of  T  the  two  lines  through  it  belonging  to  F9,3  must  be  the  two 
invariant  lines  through  the  point,  since  otherwise  at  least  one  of  them  would  be 
transformed  into  a  line  not  in  R9,3.  The  other  two  vertices  of  the  triangle  must 
be  the  two  other  points  on  these  lines  which  do  no*-  belong  to  F9,3.  Since  the  line 
joining  these  two  points  is  a  line  of  F9,3  the  points  of  F9,3  are  three  by  three  on 
the  sides  of  the  triangle  and  hence  T  and  T2  leave  F9,3  invariant.  Since  but  four 
such  invariant  triangles  can  be  selected  from  the  twelve  points  of  R9,3  there  are 
eight  and  but  eight  collineations  of  type  L  which  leave  F9,3  invariant. 

No  transformation  of  type  l1  of  period  five  or  fifteen  can  leave  F9,3  invariant 
on  account  of  its  period.  A  collineation  of  type  I3  of  period  three  is  an  homology. 
If  an  homology  H  leave  F9,3  invariant  it  can  not  have  a  point  of  F9,3  for  center 
since  on  one  of  the  invariant  lines  a  point  of  F9,3  would  be  transformed  into  a 
point  of  R9,3.  If  any  point  P  of  R9,3  can  be  the  center  of  H,  through  P  pass 
two  and  but  two  lines  of  F9,3  and  the  axis  of  H  must  be  the  line  /  joining  the  two 
points  of  R9,3  which  lie  on  these  two  lines.     Since  /  contains  the  other  three  points 


For  example,  if  0  be  chosen  as  the  pomt,  the  elation  must  be       px2'=r#3 

px/—ixa 


COLLINEATION    GROUPS  OF    PG(2,2*).       *  25 

of  F0>3  a  homology  having  P  for  center  and  /  for  axis  must  leave  F9,3  invariant. 
For  each  of  the  twelve  points  of  R0,3  there  are,  then,  two  and  but  two  homologies 
leaving  F0,3  invariant.  Accordingly,  there  are  twenty-four  homologies  leaving 
FH,3  invariant. 

If  the  homology  H  having  P  for  center  and  /  for  axis  be  multiplied  by  an 
elation  E  leaving  F9,3  invariant  and  having  some  point  Q  on  /  for  center  a  col- 
lineation  T2  is  obtained  which  transforms  F0,3  into  itself  and  leaves  invariant  the 
points  P  and  Q  and  the  line  /.  Since  T2  is  of  determinant  not  unity  and  is  of 
period  2  on  /  and  period  3  on  the  line  PQ  it  must  be  a  collineation  of  type  II. 
Since  there  are  three  and  but  three  choices  for  the  point  Q  on  /  there  are  six  and 
but  six  collineations  of  type  II  having  P  for  center  which  leave  F9,3  invariant. 
But  every  collineation  of  type  II  is  of  period  six  and  has  for  its  square  a  homology 
and  for  its  cube  an  elation.  Hence  every  collineation  of  type  II  which  leaves  F9„ 
invariant  must  be  the  product  of  a  homology  and  an  elation  each  leaving  F9,3 
invariant  and  therefore  related  as  were  PI  and  E  in  obtaining  T,  above.  Since 
the  only  points  which  can  be  the  centers  of  homologies  leaving  F9,3  invariant  are 
the  twelve  points  of  R9,3  and  each  homology  and  its  square  can  be  combined  with 
three  different  elations  there  are  seventy-two  and  but  seventy-two  collineations  of 
type  II  leaving  F9,3  invariant. 

If  F9,3  can  be  left  invariant  by  a  collineation  T3  of  type  III,  T3  must  have  the 
same  center  and  axis  as  some  elation  since  T32  is  an  elation.  Taking  0  for  center 
and  0  4  for  axis  we  find  that  there  are  six  and  but  six  collineations  T3,T3',  T3", 
and  their  cubes,  of  type  III  which  leave  F9,3  invariant  and  have  this  center  and 
axis.  This  corresponds  to  the  fact  that  there  are  only  three  ways  in  which  the  four 
lines  of  F0,3  through  0  can  be  interchanged  by  pairs.  The  transformations  T3, 
T/,  and  T3"  are 

px1'=x1-\-Px2-\-ix3  px/=x1-\-ix2-{-x:i  p*i'='*i-f-,J(,:.'-|-A:3 

T3:    p*/=iAr1+/Ar2-|-/jf3  17/ :px2,=i2x1-{-ix2-\-ix3    T3":  px./=ixl-\-x.,+ix3 

px/=i2x1-\-x2-{-ix3  px/=x1-\-x2-\-ix3  px3'=rx1-\-x2-\-x3 

That  these  collineations  leave  F9,3  invariant  is  more  readily  seen  when  they  are 
written  in  the  form  (points  of  F„,3  in  italics)  : 

T,— (o)  (ill  IS  7)  {6  iS  19  io)  (1  16)  (4  14)  (5  12  9  17)  (2  13  8  20) 
T3'={o)  (J  19  15  6)  (7  io  ii  18)  (1  14)  (4  16)  (5  13  9  20)  (2  17  8  12) 
T,"—  (o)  (3  18  15  io)    (6  7  19  u)    (1  4)   (14  16)   (5  2  9  8)   (12  20  17  13) 

Since  these  collineations  are  not  commutative  with  the  collineations  of  type  I3 
and  the  elations  which  change  the  point  0  into  points  on  the  other  sides  of  the 
triangle  of  reference  it  is  clear  that  the  total  group  of  collineations  leaving  F0,s 
invariant  must  contain  six  collineations  of  type  I3  for  each  point  of  FyI,  or  alto- 
gether 54  such  collineations.  In  fact,  it  is  obvious  that  if  any  other  center  and  axis 
than  0  and  0  4  respectively  had  been  selected  six  and  but  six  collineations  of  type 
III  leaving  F9,3  invariant  and  having  that  center  and  axis  could  have  been  deter- 
mined, provided  that  the  center  and  axis  selected  were  the  center  and  axis  of  some 
elation  leaving  F9,3  invariant. 


T 

x  05 

T 

A  0 

X0 

r#2 

i2x3 

X3 

*i 

X 

26  U.  G.  Mitchell:  Geometry  and 

No  transformation  of  type  I0  of  period  21  or  7  can  leave  F9,3  invariant  on  ac- 
count of  its  period.  If  a  transformation  T0  be  of  type  I0  of  period  three  it  per- 
mutes all  points  of  the  plane  by  triangles.  As  we  have  seen,  there  are  four  trian- 
gles each  of  which  has  for  vertices  points  of  R9,3  and  all  other  points  on  its  sides 
points  belonging  to  F0,3.  If  T0  leave  F9,3  invariant  it  must  transform  this  set  of 
triangles  into  themselves,  either  by  permuting  the  vertices  of  one  of  the  triangles 
among  themselves  or  by  transforming  one  triangle  wholly  into  another.  Selecting 
one  of  these  triangles,  say  4  12  17,  we  determine  all  the  transformations  of  type 
10  of  period  three  which  permute  its  vertices  in  the  order  (4  12  17)  and  find  that 
there  are  the  six  following: 

T     T     T     T 

-"■01      ±  02      ■'■OS      x  0- 

pxi=  x2      x2     ix2      x2 

fiX2   ==  XZ        1X3  X3  X3 

px/=ix1      x1      x±     ix1 
Writing  these  in  the  form  (points  of  F0,3  in  italics) 

T01=(412  17)  (019  u)  (3  IS  io)   (6187)    (12  9)   (5  13  8)   (14  20  16) 
T02=(4  12  17)  (o  7  75)  (3  11  18)  (6  10  19)    (1  16  5)  (2  14  13)  (8  9  20) 
T03=(4  12  17)  (0  3  6)  (7  11  10)  (is  18  19)  (1  8  14)   (2  5  20)   (9  13  16) 
T04=(4  12  17)   (0  19  IS)  (3  6  10)   (11  18  7)  (114  2)   (8  13  9)   (5  16  20) 
T05=(4  12  17)  (o  3  a)  (6  18  19)  (7  15  io)(l  9  16)  (8  20  14)  (5  2  13) 
T06=(4  12  17)  (07  6)  (3  15  18)  (11  10  19)  (15  8)  (2  20  9)   (1314  16) 

it  is  seen  that  the  vertices  of  no  one  of  the  triangles  2  8  16,  5  9  14,  1  20  14  are 
permuted  by  any  of  these  transformations  but  that  one  triangle  is  transformed 
wholly  into  another.  Since  the  squares  of  these  transformations  also  leave  F9,3 
invariant  there  are  twelve  transformations  of  type  I0  of  period  three  leaving  F9,3 
invariant  which  permute  the  vertices  of  a  given  one  of  the  four  triangles  whose 
vertices  are  in  R9,3.  There  are,  therefore,  altogether  48  collineations  of  type  I0  of 
period  three  which  leave  F9,3  invariant. 

Summarizing,  we  have  in  the  total  group  leaving  F9,3  invariant  9  elations,  8  type 
I3's,  54  type  Ill's,  24  homologies,  72  type  II's,  48  type  I0's  of  period  3,  and  the 
identity,  making  a  total  of  216  collineations.  The  group  is  readily  identified  as 
the  Hessian  group  G21C  discovered  by  C.  Jordan  in  J  878.*  Here  the  9  points  of 
F9,3  represent  the  9  points  of  inflection  of  each  cubic  of  the  pencil 

M*l3+  *23+  *88)+/**l*2*S=0 

which  is  invariant  under  every  collineation  of  the  group.  To  verify 
that  every  point  of  F9,3  lies  on  every  cubic  of  the  pencil  it  is  only  necessary 
to  notice  that  every  point  of  F9,3  has  one  and  but  one  coordinate  zero  and 
z3=(r)3=l.      Every  subgroup  of  G210  leaves  F9,3  invariant  and  every  group  in 


*  Crelle,  Vol.  84,  (1878)  pp.  89-215.     The  group  is  defined  by   leaving   invariant  a  pencil 
of  cubics  XF-4-//.H^O  where  F  is  a  ternary  cubic  and  H  its  Hessian  covariant. 


Collineation  Groups  of  PG(2,22).  27 

PG(2,22)    (except   the  G21(1  itself)    which   leaves   F,„   invariant   must   be  a   sub- 
group of  the  G216.* 

(e) 


Q        3    1' 

■.  =F0,S'.  Consider  a  point   P  of   F„s'  and   the  three  lines  /,./..,/.,  of 

-J— --___' 
F0)3'  which  pass  through  P.  On  /,,/.,,/.,  are  7  points  of  F„, .'  and  there  are,  there- 
fore, two  points  P'  and  P"  of  F0,z'  not  on  /,,/.,,  or  /.,.  There  are  two  possibilities 
only — P'  and  P"  are  or  are  not  collinear  with  P.  If  P'  and  P"  are  collinear  with 
P  the  configuration  is  F9,a  except  that  ?■  lines  are  omitted.  If  P'  and  P"  are 
not.  collinear  with  P  there  must  be  two  lines  through  P  (namely  PP'  and  PP") 
which  contain  but  two  points  of  Fn,.,'  each.  Since  P  can  be  transformed  into  any 
other  point  of  F<„./  this  would  necessitate  the  existence  of  a  configuration  of  9 
points  arranged  two  points  to  the  line  which  contradicts  the  condition  that  when 
the  number  of  points  exceeds  G  there  must  be  at  least  3  to  the  line.  Hence  there 
is  but  one  possible  arrangement  and  that  is  as  the  9  points  and  9  of  the  lines  of 
F8,3.  Since  F9,3  includes  all  lines  joining  its  points  it  follows  that  every  trans- 
formation which  permutes  the  9  points  and  9  of  the  lines  of  F9,3  also  permutes 
«imong  themselves  the  other  3  lines  of  F9,a.  F,„3'  is,  therefore,  the  subgroup  GM  of 
order  54  of  F9,3  which  leaves  invariant  a  simple  3-line  composed  of  3  lines  of  F0,, 
jo  chosen  that  no  two  of  them  meet  in  a  point  of  F9,3. 

j  3      8  I  =Irs>3'  Let  any  line  of  F8,3  be  chosen  as  the  line  x3=o.     Since 


but  6  lines  of  F8,3  meet  x3=0  in  points  of  F8,3  there  is  one  and 
jut  one  line  of  F8,3  which  meets  .v3=o  in  a  point  of  R8,3.  Let  this  be  chosen  as 
the  line  x.,=o  and  let  x{^=o  be  the  line  joining  the  two  points  on  x.i=o  and  x2=o 
(other  than  their  point  of  intersection)  which  belong  to  RK,3.  The  points  3,  7, 
19  of  xz=o  and  6,  11,  15  of  x2=o  are  then  points  of  F8,3.  Through  each  of  the 
points  3,  7,  19  pass  3  lines  of  F8,3  which  contain  7  points  of  F8,:!.  Since  the  one 
other  point  determines  but  one  line  with  a  given  point  it  follows  that  through  each 
of  the  points  3,  1,  19  passes  one  and  but  one  line  which  contains  no  other  point  of 
F8,3.  These  3  lines  must  meet  x.2=o  in  a  point  of  R8,3  and  hence  all  pass  through 
the  point  12  the  intersection  of  x.2=o  and  *1=o.  Any  line  through  4  ( i,o,o)  other 
chan  x2=o  and  x3=o  intersects  these  three  lines  in  points  of  R8,3  and  hence  can 
contain  no  points  of  F8,3  except  as  points  of  intersection  with  at,=<7.  Therefore  the 
other  two  points  of  Fs,3  lie  on  xt—0.  But  we  know  that  the  9  points  other  than 
vertices  on  the  sides  of  any  triangle  in  PG(2,2-)  are  collinear  by  threes  on  12  lines 
of  which  4  pass  through  each  point.  Accordingly,  if  any  one  of  the  points  0,  10, 
18  on  xx=o  be  omitted  there  remain  8  points  collinear  by  threes  on  8  lines.     Hence 

*  The  group  G216  was  studied  at  length  by  Maschke,  Math.  Annulen,  Vol.  33  (1890),  pp. 
324-330,  and  the  geometric  properties  of  the  group  and  its  subgroups  by  Newson,  Kansas  Uni- 
versity Quarterly,  Vol.  II,  No.  6  (Apr.  1901),  pp.  13-22. 


28  U.  G.  Mitchell:  Geometry  and 

F8,3  must  be  F9,3  with  one  point  and  the  4  lines  through  that  point  omitted  and  its 
group  is  the  subgroup  of  F9,3  G24  of  order  24  which  leaves  invariant  a  single  point. 

<e)  pf — Q-r 

•=F7,3.  Let  /  be  any  line  of  F7,3  and  P,  Q,  R  the  three  points  on 


/  belonging  to  F7,3.  Then  there  are  but  four  other  points  A,B,C,D  0f  F7,s 
not  on  /  and  these  four  points  must  lie  two  by  two  on  two  lines  through 
each  of  the  points  of  the  complete  quadrilateral  of  the  other  four  points  A,B.C,D. 
Let  A,B,C,  be  taken  (See  Fig.  1,  p.  4)  as  the  vertices  (7,0,0), (0,0,1)  (0,1,0) 
respectively  of  the  triangle  of  reference  and  D  as  the  point  (/,/,/).  This  deter- 
mines the  coordinates  of  P,Q,R  as  ( 1,0,1), (0,1, 1),  (1,1,0)  respectively.  Since 
these  coordinates  are  in  the  GF(2)  F7,3  coincides  with  PG(2,2)  and  we  can  de- 
termine the  number  of  collineations  of  each  type  by  substituting  n=l  in  the 
formulae  given  on  p.  10,  noting  that  type  \x  of  PG(2,2)is  type  I3  of  PG(2,22). 
This  gives  56  collineations  of  type  I3  (type  \x  in  PG(2,2)  ),  48  of  type  I0  of 
period  7,  42  of  type  III,  and  21  elations,  which,  together  with  the  identical  trans- 
formation make  a  group  G1G8  of  order  168.  Every  collineation  in  the  group  is 
of  determinant  unity  and  since  it  is  of  degree  7  and  order  168  it  is  recognized  as 
the  simple  group  G]68  first  derived  by  Klein  by  the  consideration  of  the  trans- 
formation of  the  seventh  order  of  elliptic  functions.* 

Every  group  which  leaves  F7,3  invariant  must  be  a  subgroup  of  G1G8  and  every 
subgroup  of  the  G]G8  leaves  F7,3  invariant! 


(f)  r«-^  (g) 


6     5  I  =F     • 
2     15  c'2' 


6 

4 

2 

12 

6 

2  1 

2 

• 

(h) 


6      3 


F     '.  o      0     _F     ' 

°'2  '  2      9  6'2 


(i) 

=F 

Since  the  six  points  of  F6,2  are  no  three  collinear  we  may  choose  any  three  of 
them  for  the  vertices  4  (1,0,0),  12  (0,0,1),  17  (0,7,0)  of  the  triangle  of  reference 
and  any  point  not  collinear  with  any  two  of  these,  say  (l,I,i)  may  be  taken  as  the 
fourth  point.  Since  in  PG(2,22)  the  choice  of  four  points  no  three  of  which  are  col- 
linear determines  uniquely  (Cf.  Corollaries  4  and  5  of  Theorem  4)  the  other  two 
points  not  collinear  with  any  two  of  them,  the  fifth  and  sixth  points  are  neces- 
sarily 13  (i,i,i)  and  20  (ijj).     Six  points,  no  three  of  which  are  collinear,  de- 

*  Math.  Annalen,  Vol.  14  (1878),  p.  438.  Jordan,  in  determining  the  finite  ternary  groups 
missed  both  this  group  and  the  simple  G36O.  The  G168.  is  discussed  at  some  length  by  Burnside, 
Theory  of  Groups,  pp.  208-209  and  302-305,  and  in  Klein-Fricke's  Modulfunctionen. 

■f  For  a  list  of  all  groups  whose  degree  does  not  exceed  8,  see  Miller,  Amer.  Journal  of  Math, 
Vol.21  (1899),  p.  326.  The  types  of  substitutions  and  of  subgroups  of  the  G168  are  given  by  Gor- 
dan,  Math.  Annalen,  Vol.  25,  (1885),  p.  462. 


COLLINEATION   GROUPS   OF    PG(2,22).  29 

termine  15  distinct  lines.  Hence  every  collineation  which  leaves  invariant  the 
six-point  also  leaves  invariant  the  associated  fifteen-line.  From  this  is  follows 
that  the  groups  leaving  F,,,/,  F„,2",  F(„2 '" ,  invariant  must  either  be  the  group 
leaving  F0,2  invariant  or  subgroups  of  it. 

An  elation  E  leaving  Fc,2  invariant  can  not  have  more  than  two  points  of  F0„ 
on  its  axis,  and  since  an  elation  interchanges  by  pairs  all  points  not  on  its  axis  K 
must  interchange  by  pairs  at  least  four  points  of  F,„2.  Since  any  four  points  no 
three  of  which  are  collinear  can  be  transformed  into  any  four  such  points  by  a 
collineation  there  exist  in  PG(2,2-)  collineations  interchanging  by  pairs  any  four 
points  of  Ftl,2.  All  such  collineations  are  elations  because  no  other  transforma- 
tions in  PG(2,22)  are  of  period  two.  Moreover,  such  an  elation  E  leaves  inva- 
riant each  diagonal  point  of  the  complete  quadrangle  of  the  four  points  chosen 
and  therefore  the  other  two  points  on  the  diagonal  line  which  are  not  diagonal 
points.  These  last  two  points  must  be  the  other  two  points  of  F„,2  (Cf.  proof  of 
Cor.  5,  Theorem  4).  Since  four  points  no  three  of  which  are  collinear  can  be 
interchanged  by  pairs  in  three  different  ways  it  follows  that  there  are  three  ela- 
tions leaving  F0,2  invariant  for  each  distinct  quadrangle  that  can  be  chosen  from 
F0,2.     There  are  then  3(0',;*>'4*3/4!)=4f>  elations  which  leave  F„2  invariant. 

Since  any  four  points  no  three  of  which  arc  collinear  can  be  transformed  into 
any  four  such  points  by  a  collineation  there  exist  in  PG(2,2-')  collineations  per- 
muting any  four  points  of  F„,2  in  any  given  cyclic  order.  Such  a  transformation 
must  be  of  period  four  and  therefore  of  type  III.  A  collineation  T  of  type  III 
which  permutes  in  cyclic  order  any  four  points  of  F,.,2  must  leave  invariant  one 
of  the  diagonal  points  of  the  complete  quadrangle  of  the  four  points  and  inter- 
change the  other  two  diagonal  points.  Since  any  two  points  of  F„,2  lie  on  the 
diagonal  line  of  the  complete  quadrangle  of  the  other  four  points  but  are  not 
diagonal  points  it  follows  that  if  T  permutes  in  cyclic  order  four  points  of  F„,, 
it  interchanges  the  other  two  points  of  Fa>2.  Since  any  four  points,  no  three  of  which 
are  collinear,  can  be  permuted  in  six  different  cyclic  orders  it  follows  that  there 
are  6  (6*5 4 -3/4!)  =90  collineations  of  type  III  which  leave  F„.2  invariant. 

Since  a  collineation  of  type  lx  of  period  5  leaves  invariant  one  real  point  if  it 
leave  Ffl,2  invariant  its  center  must  be  a  point  of  F(!,2.  The  other  points  of 
Fe,a  form  a  conic  of  which  the  sixth  point  (the  center)  is  the  outside  point.  If 
4  (i,o,o)  be  taken  as  the  center  and  the  other  five  points  permuted  in  the  order 
(1  13  20  12  17)  the  transformation  is  found  to  be 

T:     px./=rx.,-\-x:i 

of  type  Ij  and  period  5.  It  was  shown  in  §  3  that  there  are  six  different  pairs  of 
imaginary  points  on  the  axis  of  T  determining  six  transformations  of  type  I,  with 
the  same  center  and  axis  and  no  one  a  power  of  another.  These  correspond  to  the 
six  different  cyclic  orders  in  which  five  points  of  the  conic  can  be  permuted  and 
there  are,  therefore,  G  independent  transformations  of  period  5    (24  altogether) 


30  U.  G.  Mitchell:  Geometry  and 

leaving  FG,2  invariant  and  having  the  point  4  for  center.  Hence  there  are  alto- 
gether G -24=144  collineations  of  type  I±  leaving  FG,2  invariant. 

Since  a  transformation  of  type  I3  permutes  in  cyclic  order  any  set  of  three  points 
so  related  to  the  vertices  of  its  invariant  triangle  that  the  six  points  are  no  three 
collinear,  any  collineation  of  type  I3  which  has  three  points  of  F6,2  for  vertices 
of  its  invariant  triangle  must  leave  F0,2  invariant.  Twenty  distinct  triangles  can 
be  chosen  from  the  six  points  of  F0,2  and  hence  40  collineations  of  type  I3  leave 
F0,2  invariant  in  this  way.  We  know,  however,  that  each  such  collineation  per- 
mutes the  vertices  of  two  triangles  not  in  Fr>,2  either  of  which  may  be  taken  as 
the  invariant  triangle  of  a  transformation  of  type  I3  which  permutes  the  vertices  of 
the  two  triangles  in  Fp,2.  For  each  pair  of  triangles  that  can  be  selected  in  Fc,2 
there  are  then  four  transformations  of  type  I3  having  invariant  triangle  not  in 
FG,2  which  leave  F0,2  invariant.  Since  ten  pairs  of  triangles  can  be  chosen  in  FG,2 
Lhere  are  40  collineations  of  type  I3  which  leave  Fc,2  invariant  in  this  way. 

It  has  been  shown  that  the  group  which  leaves  FG,2  invariant  must  contain 
as  many  as  360  collineations.  Since  the  group  can  be  represented  as  a  substitu- 
tion group  on  6  symbols  it  must  therefore  be  either  the  alternating  or  symmetric 
group  of  degree  six.  But  since  there  is  no  collineation  which  holds  four  of  the 
points  of  F6,2  each  fixed  and  interchanges  the  other  two  the  group  can  not  be  the 
symmetric  group.  The  group  which  leaves  F0,2  invariant  is  therefore  the  alternat- 
ing group  on  six  symbols,  shown  by  Wiman*  to  be  identical  abstractly  with  the 
the  finite  ternary  group  G300  first  set  up  bv  H.  Valentiner.f 

The  group  G3fi0  is  here  characterized  not  only  as  a  group  on  six  points  but 
since  any  one  of  the  points  can  be  taken  as  the  outside  point  of  the  conic  deter- 
mined by  the  other  five  points  as  a  group  leaving  invariant  a  system  of  six  conies. 
It  is  clear  that  every  subgroup  of  G3(i0  leaves  F,.,2  invariant  and  every  group  in 
PG(2,22)  which  leaves  Fr>,2  invariant  must  be  a  subgroup  of  G3fi0. 


0) 


5      4 
2     10 


=F„2; 


(k) 


5      2 
2      5 


Fn,/. 


Since  the  five  points  of  F-,2  are  no  three  collinear  they  form  a  conic  and  the 
group  which  leaves  Fa,2  invariant  is  therefore  simply  isomorphic  with  the  group 
cf  all  transformations  of  points  on  a  line.  The  group  accordingly  contains  15 
elations,  20  type  I3's  and  24  type  I/s  of  period  5  (Cf.  §  3).  Since  every  trans- 
formation which  leaves  a  conic  invariant  must  leave  its  outside  point  invariant 
this  group  is  recognized  as  the  subgroup  of  the  G3no  which  leave?  a  single  point 
fixed.  It  is  here  represented  joth  as  the  group  which  leaves  invariant  a  conic  and 
the  alternating  group  on  five  symbols  (the  five  points  of  F-,2).  The  group  which 
leaves  F5,2'  invariant  must  be  the  subgroup  of  the  G„0  which  leaves  the  10  lines  of 

*  Math.  Annalen,  Vol.  47,  (1896),  p.  531. 

f  Kjoeb.  Skr.  (5)  5  (1889),  p.  64.  See  Ency.  d.  Math.,  ffiss.,  Vol.  I,  p.  529.  In  deter- 
mining the  finite  ternary  groups,  Valentiner,  who  was  apparently  unaware  of  the  previous  work  of 
Klein  and  Jordan,  missed  the  G36O. 


COLLINEATION   GROUPS  OF    PG(2,2»).  U 

FB>a  invariant  in  two  systems  of  five  lines  each.     It  is  therefore  a  group  G10  of 
order  10. 

(I)pr"M_F  .  <m>j  i    ,|_F  , 

|_2 0J     ^4'l■,,  LL_LL         ' 

The  configuration  F4,2  is  a  complete  quadrangle  and  since  four  points  can  be 
permuted  among  themselves  in  all  ways  by  transformations  in  the  plane  the  group 
must  be  the  symmetric  group  G21  of  all  transformations  on  the  four  points.  The 
configuration  F4,3'  is  a  simple  quadrangle  and  hence  its  group  is  the  subgroup  Gc 
of  the  G,,. 


<»>  rt-r 

I  2      3 


=F3,2.  The  configuration  F„a   is  the  triangle  and   hence  every 


transformation  leaving  it  invariant  must  do  so  in  some  one  of  the  following  three 
ways:  (1)  Leave  the  three  vertices  each  fixed;  (2)  leave  one  vertex  fixed  and 
interchange  the  other  two;  (3)  permute  the  three  vertices  in  cyclic  order. 

These  may  be  written  down  at  once  as  follows: 

Under  (1)  there  are  2  type  I3's  and  G  homologies;  under  (2)  there  are  18 
type  II's  and  9  elalions;  under  (3)  there  are  (>  type  I.,'s  and  12  type  I,,'s  of  period 
3.     Including  the  identity,  then,  there  are  54  collineations  in  the  group. 

§  8.  Subgroups  of  the  Group  G2880  Which  Leaves  a  Line  Invariant. 

All  groups  which  leave  invariant  a  set  of  collinear  points  must  also  leave  in- 
variant the  line  /  which  contains  the  points.  Since  any  four  lines  no  three  of  which 
are  concurrent  can  be  transformed  into  any  four  such  lines  by  a  projective  trans- 
formation in  PG(2,pn)  the  order  of  the  group  leaving  a  line  fixed  is 
N=  (/,=«+/,"  )(^»)^"— 2/>"4-l)=/r"(AJ"— 1  )(/>"—  1) 

For  PG(2,22)  this  gives  N—2880  and  accordingly  the  group  will  be  desig- 
nated as  G2880.  Since  any  line  in  the  plane  can  be  transformed  into  any  other  line 
in  the  plane  by  a  collineation  within  the  G0048o  it  follows  that  G,;„1H„  contains  21 
conjugate  groups  G2880. 

Subgroups  of  G..880  which  lean-  a  point  not  on  I  invariant.  In  determining  the 
subgroups  of  G2880  we  shall  first  determine  all  subgroups  which  leave  invariant  at 
least  one  point  not  on  /  and  then  all  subgroups  which  leave  invariant  no  point  not  on 
«\     Taking  /  as  the  line  *3=<>   (or  the  line  at  infinity)   every  collineation  in  the 

G2(n    is  of  the  form 

x'=alx-\-a.,y-\-a 
l!    y'—btx+bj+b 
where  x   and  y   are  nonhomogeneous  point  coordinates.      Selecting   the   point   not 
on  /  as  tie  origin  every  collineation  in  the  G.,s„„  which   leaves  it   invariant  is  of 

the  form 

x'=alx-{-a.,y 
1  <  :  y'=bxx+b,y 
But  it  has  been  seen  in  §  3  that  in  homogeneous  coordinates  the  group  of  all  trans- 


32  U.  G.  Mitchell:  Geometry  and 

formations  of  the  form  T0  is  the  group  Geo  of  all  transformations  of  points  on 
.1  line.  Since  Gco  is  the  alternating  group  on  five  symbols  it  contains  the  follow- 
ing subgroups:* 

I.  Subgroup  leaving  all  points  of  the  line  fixed. 

1  self-conjugate  Gx — the  identity.     Tl :  #'=«!#,  y'-.=a1y. 

II.  Subgroups  leaving  invariant  two  points  of  the  line. 

a.  Those  leaving  each  point  of  the  pair  fixed. 

10  groups  G3  each  conjugate  to  the  G3  of   transformations  of  the   form 
T2:x,=a1x,  y,=b2yi  ay  b.,  in  the  GF(2-). 

b.  Those  leaving  the  pair  invariant. 

10  groups  G6  each  conjugate  to  the  G6  of  transformations  of  the  form  T0 
subject  to  the  restriction  that  either  a1=b2=o  or  a.,=b1=o. 
10  groups  G2  each  conjugate  to  the  subgroup  of  G6  for  which  the  coeffi- 
cients are  in  the  GF(2). 

III.  Subgroups  leaving  one  point  of  the  line  invariant. 

15  groups  G2  each  conjugate  to  the  G2  of   transformations   of  the   form 
T3:  x'=a1x-\-a2yJ  y'=  ax  y,  where  tf^^are  in  the  GF(22). 
5  groups  G4  each  conjugate  to  the  G4  of  transformations  of  the  form  T3 
where  alt  a2  are  in  the  GF(22). 

5  groups  G12  each  conjugate  to  the  G12  of  all  transformations  of  the  form 
T4:  x'=a1x-\- a2y,  y'=bs  y,  where  a1,a2b2  are  in  the  GF(22). 

IV.  Subgroups  leaving  invariant  a  pair  of  imaginary  points  on  the  line. 

a.  Those  leaving  each  point  of  the  pair  fixed. 

6  groups  Gr>  each  conjugate  to  the  G-  of  all  transformations  of  the  form 
T0  subject  to  the  condition  that 

a1--]-ia22-{-i2b12-{-b22-\-a1a2  -\-i'-a1b1-\-ra2b1-\-a2b2-\-rb1b2=o 

b.  Those  leaving  the  pair  invariant. 

6  groups  G10  each  conjugate  to  the  G10  of  all  transformations  of  the  form 
T0  subject  to  the  condition  that 

If  T0  be  taken  as  a  transformation  in  nonhomogeneous  coordinates  we  have  a 
group  G60  simply  isomorphic  with  G605  or  a  group  G180  triply  isomorphic  with 
G005  according  as  the  determinant  of  the  group  of  transformations  of  the  form  T3 
is  unity  or  unrestricted  within  the  GF(22).  Corresponding  to  the  above  groups  on 
the  line  there  are  then  the  following  groups  in  the  plane  which  leave  invariant  the 
line  /  and  point  (o,o). 

I.  Subgroups  leaving  all  points  of  /  fixed. 

1.  Of  determinant  unity,  1  self-conjugate  Gv 

2.  Of  determinant  not  restricted,  1  self -conjugate  G3. 

II.  Subgroups  leaving  invariant  a  pair  of  points  on  /. 
1.  Those  leaving  each  point  of  the  pair  fixed. 

a.  Of  determinant  unity,  10  conjugate  G3  each  leaving  a  triangle  invariant. 

b.  Of  determinant  not  restricted,  10  conjugate  G9  each  leaving  a  triangle  in- 
variant. 


All  groups  of  degree  less  than  6  were  obtained  by  Serret. 


COLLINEATION    GROUPS  OF    PG(2,22).  3J{ 

2.  Those  leaving  the  pair  of  points  invariant. 

a.  Of  determinant  unity, 

10  conjugate  G„  each  leaving  a  triangle  invariant. 

b.  Of  determinant  unrestricted, 

10  conjugate  G18  each  leaving  a  triangle  invariant. 
III..  Subgroups  leaving  one  point  of  /  fixed. 
1.  Of  determinant  unity, 

15  conjugate  G2  each  leaving  invariant  a  point  of  lines. 
5  conjugate  G«  each  leaving  invariant  a  point  of  lines. 
5  conjugate  GJ2  each  leaving  invariant  the  line  /,  the  Ar-axis  and  the  origin 
IV.     Subgroups  leaving  invariant  a  pair  of  imaginary  points  on  /. 

1.  Subgroups  leaving  each  point  of  the  pair  fixed. 
6  conjugate  G„  of  determinant  unity. 

6  conjugate  G]3  of  determinant  not  restricted. 

2.  Subgroups  leaving  the  pair  invariant. 
6  conjugate  G10  of  determinant  unity. 

6  conjugate  G30  of  determinant  not  restricted. 
Subgroups  of  G JSSII  which  leave  no  point  not  on  I  invariant.  In  the  discussion 
which  follows  the  term  translation  will  be  used  to  indicate  an  elation  having  the 
line  /  for  axis  and  the  term  elation  will  be  used  only  for  an  elation  whose  axis  is 
not  /.  In  determining  the  subgroups  of  G.,880  which  leave  invariant  no  point  not 
on  /  we  shall  first  determine  all  such  subgroups  containing  no  translations  and 
then  all  such  subgroups  containing  translations. 

Let  Gn  be  a  subgroup  of  G,880  which  leaves  no  point  not  on  /  invariant  and 
contains  no  translation.  If  Gn  contain  an  homology  H  having  /  for  axis  and  A 
for  center,  G„  must  contain  at  least  one  transform  H'  of  H  having  some  other 
point  than  A  for  center.  One  of  the  products  H'H  or  H'H-  is  of  determinant 
unity  and  leaves  all  points  on  /  fixed.  It  is  therefore  a  translation.  Hence  Gn  can 
contain  no  collineation  other  than  the  identity  leaving  all  points  of  /  fixed  and, 
consequently,  every  such  group  must  be  simply  isomorphic  with  the  G00. 

We  have  seen  (ante  p.  30)  that  there  is  a  group  G,!0  leaving  invariant  a  point 
conic  and  its  outside  point.  By  duality  there  is  a  G«0  leaving  invariant  a  line 
conic  and  its  outside  line.  Since  the  line  /  is  the  outside  line  of  48  different  line 
conies  there  are  48  such  groups  G60  which  leave  /  invariant. 

Every  transformation  of  the  form 

*'=*  +  * 

is  a  translation  and  the  group  of  all  transformations  of  the  form  E  where  a  and 
b  are  marks  of  the  GF(2-)  is  a  G10  leaving  every  point  on  /  fixed.  Unless 
a=b=o  E  is  of  period  2  and  hence  G]6  contains  15  cyclic  subgroups  of  order  two. 
If  a=o  and  b  be  allowed  to  take  on  all  values  in  the  GF(2S)  or  if  b=o  and  a 
be  in  the  GF(22)  a  group  of  order  4  is  obtained,  consisting  of  all  translations 
leaving  fixed  all  lines  through  a  given  point  P  on  /.  Such  a  group  will  be  desig- 
nated as  a  G4(P).  If  a  be  allowed  to  take  the  value  o  and  but  one  other  value 
and  b  be  restricted  in  the  same  way  a  group  of  order  4  is  obtained  containing  be- 
sides the  identity  3  translations  no  two  of  which  have  the  same  center.  Since  such 
a  group  leaves  invariant  a  complete  quadrangle  of  which  the  centers  of  the  three 
elations  are  the  diagonal  points,  it  will  he  designated  as  a  G,(Q).     If  a  be  in  the 


34 


U.  G.   Mitchell:  Geometry  and 


GF(22)  and  b  in  the  GF(2)  a  group  G8  is  obtained  leaving  invariant  a  point  P 
on  /  and  interchanging  the  four  lines  other  than  /  through  P  by  pairs  in  a  given 
manner. 

The  G16  is  an  Abelian  (or  commutative)  group  since  if  Ei  and  Ej  be  any  two 
clations  in  G18  EjEi=E1Ej.  Consequently  EjE1Ej-1=EiEjEj-1=E1,  and  every 
subgroup  of  G16  is  self-conjugate  within  the  G1C.  Also  every  two  translations  and 
their  product  form  (with  the  identity)  a  group  G4,  for  if  EiEj=Ek, 
E1=EkEj=EjEk  and  Ej=EkE1=EiEk.  Hence  G10  contains  15-14/0=35  sub- 
groups G4.* 

We  shall  next  determine  the  subgroups  of  G2880  which  are  such  that  every  col- 
lineation  in  the  group  either  leaves  invariant  a  point  not  on  /  or  is  the  product  of 
such  a  collineation  and  a  translation  in  the  group. 


*  It    may    be   of  interest   to  note  that  the  Gi6  can  be  represented  as  a  three-space  PG<3,2)  by 
letting  the  G2,  G4,  G$,  correspond  to  the  points,  lines  and  planes,  respectively,  of  the  three-space. 


The  three-space   S3  has 

The  Group  Gi6  has 

15    points,    3  5   lines,  15  planes, 

15  subgroups  G2,   35G4,  15  Gf, 

arranged 

arranged 

3  points  on  each  line, 

3  G2  in  each  G4, 

7  points  on  each  plane, 

7  G2  in  each  Gs, 

7  lines  through  each  point, 

7  G4  containing  each  G2, 

7  lines  in  each  plane, 

7  G4  contained  in  each  Gg, 

3  planes  through  each  line, 

3  G«  containing  eachG4, 

7  planes  through   each  point. 

7  Gs  containing  each  G2. 

If  the  three-space  S3  be  represented  by  the  notation  for  a  configuration  (Cf.  Moore,  American 
Journal  oj  Mathematics,  Vol.  18,  pp.  264-303;  Veblen  and  Young,  Projective  Geometry.  Vol.  I,  p 
38),  the  same  table  exhibits  the  structure  of  the  group  Gi6. 


In  the  table,  S0  is  a  point,  Si  a  line,  S2  a  plane,  and   the    interpretation    is    obvious   from    the 
parallelism  given  above. 


COLLINEATIOX    GROUPS   OF    PG(2,28).  35 

The  translations  in  any  subgroup  of  G2880  form  a  self -con  jugate  subgroup  Gk. 
If  any  group  G„  has  a  system  of  transitivity  S  which  is  also  a  system  of  transitivity 
of  its  self-conjugate  subgroup  Gk  of  translations  then  every  collineation  in  G„  is 
either  a  collineation  leaving  a  point  O  of  S  invariant  or  the  product  of  such  a  col- 
lineation and  a  translation ;  for,  let  O  be  taken  as  the  origin  and  let  T  be  any 
collineation  in  G„.  If  T  displaces  O  it  changes  O  to  some  point  A  in  S.  But  since 
S  is  a  system  of  transitivity  for  Gk  there  is  in  Gn  a  translation  Tt  changing  A  to  O. 
Hence  T1T=T2  a  collineation  in  Gn  leaving  ()  invariant.  From  T1T=T, 
we  have  T=T1T2.  Every  such  group  Gn  can  be  obtained  then  by  extending 
the  groups  leaving  a  point  fixed  by  means  of  translations.  In  determining  the 
groups  below,  S  will  be  used  to  indicate  the  system  of  transitivity  common  to  the 
group  obtained  and  the  extending  group  of  translations.  E,,  E2  and  E  will  be 
used  to  indicate  the  forms  of  translations  as  follows: 

l~  y'=y  ^     /=y+*  y'=y-H 

The  product  of  T0=  ^T'^V      and  E  is  of  ^  form 
y  =blX+b2y 

>Y  T?=r-x'=<?1x-\-any-\-a1a  -\-a2b 

It  should,  therefore,  be  observed  that  in  extending  a  group  of  collineations  of 
the  form  T0  by  Elf  E2  or  E  the  range  of  values  which  can  be  assumed  by  the 
additive  constants  in  the  product  is  dependent  upon  the  coefficients  of  T0  as  well 
as  those  of  Ex,  E2  or  E.  In  the  work  below  S  will  be  used  to  indicate  a  system  of 
transitivity  of  the  extended  group  Gn  which  is  also  a  system  of  transitivity  of  the 
extending  group  Gk  of  translations.  The  groups  obtained  by  such  extensions  are 
as  follows: 

I.  Subgroups  leaving  each  point  of  I  fixed. 

1.  Extensions  of  G!  by  E1,E2,E  give  groups  of  translations  only. 

2.  Extensions  of  the  G3  of  the  form  Tx:  x'=  a]x,y'=a1y,  ax  in  GF(2-;. 

a.  By  Ej   gives  a  G12   leaving  invariant  the   x-axis.     The  points  on   the 
x-axis  form  the  system  S. 

b.  By  E2  gives  a  similar  G12  leaving  invariant  the  y-axis. 

c.  By  E  gives  a  G48  leaving  invariant  no  point  not  on  /  and  no  line  but  /. 
The  system  S  includes  all  points  not  on  /. 

II.  Subgroups  leaving  a  pair  of  points  on  I  invariant. 

1.  Subgroups  leaving  each  point  of  the  pair  fixed.     (10  of  each). 

A.     Those    of    determinant    unity.      Extensions   of    G,    of    the    form   T, : 
x'=axx,  y'=  b2y,  ax  and  b2  in  the  GF(2-). 

a.  By  Ex  gives  a  G12  leaving  the  x-axis  invariant. 
The  system  S  includes  all  points  on  the  x-axis.. 

b.  By  E2  gives  a  similar  Gl3  leaving  the  y-axis  invariant. 

c.  By  E  gives  a  G4S  leaving  invariant  no  point  not  on  /  and  no  line  but  /. 
The  system  S  includes  all  points  not  on  /. 


36  U.  G.    Mitchell:  Geometry  and 

B.  Those  of  determinant  not  restricted.     Extensions  of  the  G9  of  the  form 
T2.     (10  of  each). 

a.  By  Ex  gives  a  G36  leaving  the  Ar-axis  invariant. 
The  system  S  includes  all  points  on  the  Ar-axis. 

b.  By  E2  gives  a  similar  G36  leaving  the  y-axis  invariant. 

c.  By  E  gives  a  G144  leaving  invariant  no  point  on  /  and  no  line  but  /. 
The  system  S  includes  all  points  not  on  /. 

2.   Subgroups  leaving  the   two   points  invariant  as  a  pair. 

A.  Those  of  determinant  unity.      (10  of  each). 

Extensions    of    the    G2    of    T0    with    the    form    «1==Z'2=o  or  a.^bx=o, 
a1,a2,b1,b2  being  in  the  GF(2). 

a.  By  Ej  extends  by  E  also  giving 

For  a  in  GF(2)  a  G8  leaving  invariant  a  PG(2,2). 
The  system  S  includes  four  points  not  on  /  no  three  of  which  are  collinear. 
For  a  in  the  GF(22)  a  G32  leaving  invariant  no  point  not  on  /  and  no 
line  but  /.    The  system  S  includes  all  points  not  on  /. 

b.  By  E,  gives  same  as  by  Ex  or  E. 

Extension  of  the  G0  of  the  form  T0     with  a1=b2-=o,  or  a2=b1^o  and 
a1,b1)a2Jb2  in  the  GFi22). 

By  E1  or  E2  extends  by  E  giving  a  Goe  leaving  invariant  no  point  not 
on  /  and  no  line  but  /.     The  system  S  includes  all  points  not  on  /. 

B.  Those  of  determinant  not  restricted.     (10  of  each) 

Extension  of  the  G48  by  Ex  or  E2  extends  by  E  with  a  and  b  unrestricted 
giving  a  GJS8  leaving  invariant  no  point  not  on  /  and  no  line  but  /. 
The  system  S  includes  all  points  not  on  /. 

III.  Subgroups  leaving  one  point  on  I  fixed. 
1.  Those  of  determinant  unity. 

A.  Extensions  of  G2  of  form  T4 :x'=  avx-\-a2yj  y'=a,y,  a^*  in  GF(2). 

a.  By  Ex  with  a  in  the  GF(2)  gives  a  G4  leaving  invariant  the  Ar-axis. 
(15  such  groups).  The  system  S  is  the  points  on  the  Ar-axis  having  coor- 
dinates in  the  GF(2). 

b.  By  Ex  with  a  in  the  GF(22)  gives  a  G8  leaving  the  Ar-axis  invariant. 
The  system  S  includes  all  points  on  the  Ar-axis. 

c.  By  E2  with  b  in  the  GF(2)  extends  by  E  with  a  and  b  in  the  GF(2) 
giving  a  G8  leaving  invariant  a  PG(2,2)  which  is  also  the  system  S. 

d.  By  E2  with  b  in  the  GF(22)   gives  a  G32  leaving  invariant  no  point 
not  on  /  and  no  line  but  /.    The  system  S  includes  all  points  not  on  /. 

B.  Extensions  of  G4  of  form  T4  with  alt  a2  in  GF(2-).     (5  of  each). 

a.  By  Ex  gives  a  G1G  leaving  invariant  the  Ar-axis. 
The  system  S  includes  all  points  on  the  Ar-axis. 

b.  By  E2  extends  by  E  giving  G04  leaving  invariant  no  point  not  on  / 
and  no  line  but  /.     The  system  S  includes  all  points  not  on  /. 

C.  Extension  of  G12  of  form  T5:   x'=axx  -\-  a2y,  y'=b2y.  (5  of  each). 

a.  By  Ex  gives  a  G48  leaving  invariant  the  *-axis. 
The  system  S  includes  all  points  on  the  Ar-axis. 

b.  By  E2  extends  by  E  giving  a  G64  leaving  invariant  no  point  not  on  / 
and  no  line  but  /. 

The  system  S  includes  all  points  on  the  Ar-axis. 


Collin eation  Groups  of  PG(2,2»).  .'{7 

2.  Those  of  determinant  not  restricted.    (5  of  each). 

A.  Extensions  of  G,2  of  form  T4. 

a.  By  E,  gives  a  G4S  leaving  a  point  of  lines  invariant. 
The  system  S  includes  all  points  on  the  *-axis. 

b.  By  E2  extends  by  E  giving  a  G,M  leaving  invariant  no  point  not  on 
/  and  no  line  but  /.   The  system  S  includes  all  points  not  on  /. 

B.  Extensions  of  Gsa  of  form  T6. 

a.  By  E,  gives  a  Gl44  leaving  the  #-axis  invariant. 
The  system  S  includes  all  points  on  the  *-axis. 

b.  By  E.,  extends  by  E  giving  a  GnTfl  leaving  invariant  no  point  not  on 
/  and  no  line  but  /.    The  system  S  includes  all  points  not  on  /. 

IV.  Subgroups  leaving  invariant  a  pair  of  imaginary  points  on  I.     (6  of  each). 
In  each  case  the  system  S  includes  all  points  not  on  /  and  no  point  on  /  and  no 
line  other  than  /  is  invariant. 

1.  Subgroups  leaving  each  point  of  the  pru'r  fixed. 

A.  Of  determinant  unity. 

Extension  of  Gs   of   form  T„    (subject  to  quadratic  condition)    by   E, 
or  E,  extends  by  E  giving  a  Gso. 

B.  Of  determinant  not  restricted. 

Extension  of  Gla  of  form  T0  (subject  to  quadratic  condition)   by  Ex 
or  E2  extends  by  E  giving  a  G,40. 

2.  Subgroups  leaving  the  points  invariant  as  a  pair. 

A.  Of  determinant  unity. 

Extension  of  G10  of  form  T0  (subject  to  cubic  condition)  by  E,  or  E, 
extends  by  E  giving  a  G1C0. 

B.  Of  determinant  not  restricted. 

Extension  of  G30  of  Form  T0  (subject  to  cubic  condition)  by  E,  or  E, 
extends  by  E  giving  a  G480. 

The  extension  of  the  total  group  G00  of  the  form  T0  and  determinant  unity 
by  Ex  or  E,  extends  by  E  giving  a  group  G900  of  all  transformations  of  determi- 
nant unity  in  the  G2880.  The  extension  of  the  group  G,80  of  form  T0  by  E,  or 
E2  extends  by  E  giving  the  G2880  itself. 

There  remain  to  be  determined  the  subgroups  of  G3S80  which  leave  no  point 
not  on  /  invariant,  contain  translations,  and  are  such  that  the  self -con  jugate  sub- 
group of  translations  has  no  system  of  transitivity  which  is  also  a  system  of  tran- 
sitivity for  all  other  transformations  in  the  group.  Since  the  group  G,„  of  trans- 
lations is  transitive  on  all  points  not  on  /  no  such  subgroup  can  contain  the  G,(i, 

Let  Gn  be  such  a  subgroup  of  G2S80  containing  a  G2  as  the  largest  self-conju- 
gate subgroup  of  translations.  Selecting  S:  x'=x-{-i,  y'=y  as  the  trans- 
lation in  the  G^,  and  T:  x'=axx-\- b]y-\-ci,  y'=(iix-\-b.iy-\-c,  as  the  gen- 
eral transformation  in  the  G2sS„  (cf.  p.  31)  we  have  TST'1 :  x'=x-\-al,  >•'=>•+<*•_.• 
Since  TST"'=S  we  have  a1==i,  a.>=0  and  every  transformation  in  the  G„  is  of 
the  form  T,:  x'=x-\-ay-\-b,  y'=cy-\-d.  If  r=i  T,  is  of  period  2  if  cd=o  and 
of  period  4  if  ad=i.  If  c=i  or  r  and  b=acd,  T,  is  of  period  3.,  If  r=i  or  r 
and  b  is  not  equal  to  aid,  T  is  of  period  (>.  Hence  every  transformation 
in  G„  is  of  type  II,  III,  IV  or  V.  Also,  all  transformations  of  types  III  and  V 
have  for  center  the  point  4=(//m/)  which  is  the  center  of  S.  all  homologies 
have  for  axis  some  line  other  than  /  through  4  and  for  center  some  point  ottier 
than  4  on  /,  and  all  transformations  of  type  II  have  4  as  the  point  of  intersection 


38  U.  G.   Mitchell:  Geometry  and 

of  the  two  invariant  lines.     Since  no  homology  has  /  for  axis,  there  are  but  two 
transformations    (S  and  the  identity)    in  Gn  which  leave   /  point-wise  invariant. 
Gn   is,  therefore,  at  most    (2,  1)    isomorphic  with  some  group  in  one  dimension 
leaving  a  point  on  the  line  invariant  and  its  possible  orders  are   (cf.  p.  32)   4,  8, 
12      and      24.         That      Gn      can      be      a      cyclic      G4      follows       from       the 
fact     that     every      system     of     transitivity     of     the     G2     contains     but     two 
points.       Gn     can   not     be     a    G4    whose     transformations    are    all    of     period 
2,  since  if  E  be  one  of  the  elations  in  such  a  G4,  the  G2  and  the  G4  have  the  same 
systems  of  transitivity  on  -the  axis  of  E.     If  Gu  be  of  order  8  and  contain  trans- 
formations of  period  2  only  it  must  contain  two  elations,  Ex  and  E2,  having  the 
same  axis  lt,  since  such  a  G8  must  contain  six  elations  and  there  are  but  four 
lines  other  than  /  through  4.     We  then  have  E1E2=E3  a  third  elation  having  1^, 
for  axis.     Since  E1?  E2  and  E3  have  the  same  center  and  axis  they,  together  with 
the  identity,  form  a  group  G4(P).     The  other  three  elations  in  the  G8  would  be 
SEn  SE2  and  SE3 ;  but  since  each  of  these  elations  has  the  same  systems  of  tran- 
sitivity on  lx  as  S,  G„  can  not  be  such  a  G8.    Accordingly,  if  Gn   be   of   order   8    it 
must  contain  a  cyclic  subgroup  G4  consisting  of  the  powers  of  a  transformation  U 
of  type  III  and  since  U  must  leave  /  invariant  we  have  U"^S.     Hence  U  is<  of 
the  form  U:  x'—x-\-ay-\-b,  y'=y-\-a'2  where  a  is  not  zero.     If  the  G8  contain  an 
elation     it     must     be     of     the     form     E:  x'=x-\-a1y-\-b1J     y'=y     where     ax     is 
not    zero.     If     a=ax     the     product        EU:  x'=x^\-b-\-bx-\-i,     y'=y-\-a2     is     a 
translation     different    from     S.       If    a     is      not     equal      to     a{      the      product 
EU:  xf=x-\-(a-\-a1)yJrb-\-b1-\-a1a2,    y'=y-\-a2     is     a     transformation     of     type 
III     whose     square       must      be      identical      with      S.         If      (EU)2:  x'=x-\- 
axa2-\-i,  y'=y  be  identical  with  S  we  must  have  aY(i1-\-i=i  or  a1a'2-\-i=o;  but  if 
tf1tf2-|-l=l,    a1a2=0    which    is    impossible    since    neither    a    nor  ax    can    be    zero, 
and  if  a1a2-\~i=0,  axd2=i  which  is  impossible  since  a  and  ax  are  different  marks 
of    GF(22).        Hence     if     Gn     be     of     order     8     it     can     contain     no     elations 
and  must  have  3  cyclic  subgroups  of  order  4.     Taking  Ux:  x'=x-\-a1y-\-bl,  y'"» 
y-f-tf,2  as  any  other  transformation  of  period  4  than  U  or  U3  in  the  G8  we  have 
UUX:  x'=x-\-{a-\-a1)y-\-aa2-\-b-\-b^  y'=v+fl2+<V  and  UtU:  x'=x-\-(a-\-a^  )y 
-\-a2ax-\-b-\-b1,    y'=y-\-ar~\-a12.       If    a=a^  the  product  UUj  is  a  translation  not 
in  the  G8.     Hence  a  must  be  different  from  ax  and  the  group,  if  existent,  is  not 
Abelian    and    must    be    of    the    type*    U4=l,     Ux4=l,     U2— IV,     U^U"^ 
U4-1.     Since  these  conditions  are  satisfied  by  any  two  transformations  of  the  form 
U  and  Ulf  where  a  is  different  from  ax   and  neither  a  nor  ax  is  zero,  there  are 
four  such  groups  G8  having  the  given  G2  as  the  largest  self-conjugate  subgroup 
of  translations. 

If  Gn  be  of  order  12  it  must  be  simply  isomorphic  with  the  G12  consisting  of  all 
transformations  leaving  invariant  a  point  on  the  line '(cf.  Ill,  p.  32).  This  is 
impossible  since  the  product  of  the  translation  in  Gn  and  any  homology  in 
Gn  is  of  period  6.  Hence  Gn  can  not  be  of  order  12.  If  Gn  be  of  order  24  it 
must,  by  Theorem  5,  contain  exactly  8  transformations  of  determinant  unity  and  1G 
transformations  of  determinant  i  or  r.  The  8  transformations  of  determinant 
unity  form  a  self -con  jugate  subgroup  and  hence  the  G24,  if  existent,  must  be  the 
direct  product  of  this  G8  and  a  cyclic  G3  consisting  of  the  powers  of  a  homology 
H.  It  was  shown  above  that  a  G8  whose  transformations  are  all  of  period  2  con- 
tains 3  elations  Elf  E/,  E/',  having  the  same  axis  lx  and  three  other  elations  E2, 
E3,  E4,  having  for  axes  the  three  other  lines  /2,  lz,  lA,  respectively,  through  P.     H 

*Cf  Burnside,  1.  c,  p.  88. 


COLLINEATION    GROUPS   OF    P(  J  (2,2a)  •  39 

can  not  have  /,  for  axis  since  the  G,  and  the  Ga4  would  then  have  the  same  systems 
of  transitivity  on  /,.  Also  H  can  not  have  /L„  /.,  or  /4  for  axis,  since  if  E| 
( 1—2,3,4)  be  the  corresponding  elation,  Ej  and  H  are  not  commutative  and 
HEiH1  would  be  an  elation  not  in  the  G8.  Hence  the  self-conjugate  G8  in 
the  G,4  can  not  have  all  its  transformations  of  period  2.  Accordingly,  the  G8  in 
the  G.,4  must  contain  3  cyclic  subgroups  of  order  4.  We  may  take  this  self-con- 
jugate G8  to  be  the  one  consisting  of  all  transformations  of  the  form  x'=x-\-ay-\-k, 
y'==y-^-a-  where  a  is  any  mark  of  the  GF(28)  and  k  is  any  mark  of  the  GF(2). 
The  three  transformations  in  this  G8  of  the  form  T:  x'=x-\-ay,  y'=y-f-/r  where 
a  is  not  zero  are  of  period  4  and  no  two  belong  to  the  same  cyclic  Gr  The  G.J4. 
if  existent,  must  contain  a  homology  of  the  form  H:  x'^=x-\-by-\-bic  y'=iy-\-c 
and,  therefore,  every  product  TH :  x'=x-)-(ia-\-b)y-\-(a-\-ib)f,  yf=iy-\-a':-\-c 
where  b  and  c  are  some  two  chosen  marks  of  the  GF(22).  Since  TH  is  of  deter- 
minant i  it  is  of  type  II  or  IV  and,  hence,  (TH)3:  x'=x -\-i(<fb-\-ac-\-i) .  y'  -=y 
must  be  identity  or  the  translation  S:  x'=x-\-i,  y'=y  for  every  value  of  a  different 
from  zero  in  the  GF(22).  But  there  exist  no  two  marks  b  and  c  in  the  GF(2L") 
which  make  i((fb-\-ac-\-i)=m  where  m  is  in  the  GF(2)  for  every  value  of  a  dif 
ferent  from  zero  in  the  GF(2-).     Hence,  Gn  can  not  be  of  order  24. 

Subgroups   having    a   G4(Q)  as  a  self-conjugate  subgroup.     Let  Gra  be  a  sub 
group  of  G2S80  having  a  G4(Q)   as  its  largest  self -con  jugate  subgroup  of  transla- 
tions and  such  that  the  G4(Q)  has  no  system  of  transitivity  which  is  also  a  system 
of  transitivity  of  the  Gm.     Selecting  the  G4(Q)   as  the  group  of  all  translations 
of  the  form  S:  x'=x-\-a,  y'=y-\-b,  where  a  and  b  are  marks  of  the  GF(2),  and 
T    is    the   general    collineation    (cf.    p.    31)     in     the     G.,88o     we     have     TST"1 : 
x'=x-\-aa1-\-bb1,  yf=y-\-aat-\-bbs.     Hence  if  TST"1  belong  to  the  G4(Q)  ««,-)- 
bb,=a'  and  aa.2-\-bb._.=b'  where  a'  and  b'  are  in  the  GF(2).     Since  these  equa- 
tions must  hold  true  for  every  au  a,,  />,,  b.,  in  the  Gm  no  matter  what  marks  of  the 
GF(2)  a'  and  b'  may  be  it  follows  that  ait  a2,  £,,  b.z  are  in  the  GF(2)  and  even- 
transformation  in  the  Gm  is  of  determinant  unity.     Consequently,  Gm  can   contain 
no  homology  and  is  at  most  (4,  1)  isomorphic  with  some  group  on  the  line  leaving 
invariant  a  pair  of  points.     The  possible  orders  of  Gm  are,  therefore,   (cf.  p.  32) 
12  and  24.     The  G4(Q)   leaves  each  point  on  /  (x^—o)   invariant  and  permutes 
the  other  36  points  in  four  systems  of  transitivity  each  consisting  of  four  points  no 
three  of  which  are  collinear.      These    four    quadrangles    are    Q,==(0    16    20    18) 
Q2s=(l  14  8  5)  ;  Q2=(2  13  6  15)  ;  Q4=(  9  10  12  11)    (cf.  Table  of  alignment, 
p.    3).      Every    transformation  in  Gm  is  of  the  form  T:  x'=a1x-\-b1y-\-cl,  y'= 
n.je-\-bzy-\-ci   where  ax,  a2,  bit  b2  are  in  the  GF(2),  and  must,  therefore,  be  of  type 
V  (elation,  period  2),  type  Ia  (period  3)  or  type  III  (period  4).     From  the  forms 
of  T2,    T:\    T4    it    appears    that    every    elation    in    Gm    must    be    of    the    form 
E, :  x'—y-\-k,  y'=x-\-k,  or  of  the  form  E.:  x'=x-\-y-\-c,  y'=y,  or  of  the  form 
E3:  x'=x,  y'=x-j-y-\-d ;  every  transformation    of   type    I,    must   be    of    the   form 
T, :  x'=y-\-m,  y'=x^-y-\-n,  or  of  the  form  T2:  x'=x-|-y-j-r,  y'—x-\-s;  every 
transformation  of  type  III  must  be  of  the  form    U, :  x'=y-\- k,  y'=x-\-l  where  / 
and  k  are  not  the  same  mark,  or  of  the  form  IL:  x'=x-\-y-\-cl,  y'— y-\-ct%    or    of 
the    form    U„:  x'=x-\-dly    y'=x-\-y-\-d2.      If  Gm  be  of  order  12  it  must  be  the 
direct  product  of  a  cyclic  G.,(cvc.  I8)   and  the  G,(Q).     But  each  transformation 
of  period  3  in  Gm  must  leave  one  of  the  quadrangles  Qi(i=l,  2,  3,  4)   invariant 
and  permute  the  other  three  in  cyclic  order.     Hence  some  one  of  these  quadrangles 
would  be  a  system  of  transitivity  of  the  (i,_.  generated  by  any  Ga(cyc  I3)  and  the 
G4(Q)  and  Gm  can  not  be  or  order  12.     If  Gm   be  of  order  24  it  must  contain  a 


40  U.  G.   Mitchell:   Geometry  and 

transformation  T  of  period  3  leaving  invariant  a  quadrangle  Qa  (a=l,  3,  3  or  4). 
Every  transformation  of  the  form  T1  or  T2  is  seen  to  leave  invariant  the  two 
points  7==(i1i,o)  and  19^(/,j,0)  on  /,  and  since  every  translation  in  the  G4(Q) 
leaves  every  point  on  /  invariant  the  eight  products  obtained  by  multiplying  T  and 
T2  by  the  transformations  in  the  G4(Q)  are  eight  transformations  of  period  3 
leaving  Qa  invariant.  Moreover,  G24  can  not  contain  any  transformation  Tt  of 
period  3  not  included  among  these  eight,  for  the  products  of  T  and  the  three 
other  transformations  leaving  Qa  invariant  and  making  the  same  permutation  of 
points  on  /  as  T  give  the  three  translations  in  the  G4(Q)  and  consequently  one 
of  the  products  TjT  or  TjT2  would  be  a  translation  not  in  the  G4(Q).  Hence, 
the  G4  must  contain,  besides  the  G4(Q),  8  transformations  of  type  I.{  leaving  Qa 
invariant  and  some  transformation  Sa  of  period  2  or  4  transforming  Qa  into  one 
of  the  other  quadrangles.  Let  S}  be  of  period  2  or  4  interchanging  the  four  quad- 
rangles in  any  order  R1=(QaQb)  (QcQd)  where  a,  b,  c,  d  are  the  numbers 
1,  2,  3,  4  in  an  arbitrary  order.  Since  half  of  the  transformations  of  period  B  in 
the  G24  must  make  the  transformation  R2=(Qa)  (QbQcQd)  on  the  four  quad- 
rangles Qi(/=1,  2,  3,  4)  the  G24  would  then  contain  a  transformation  of  period  3 
making  on  Qt(t— 1,  2,  3,  4)  the  transformation  R2R,=  (QaQbQc)  (Qd)  which 
has  just  been  shown  to  be  impossible.  Hence  the  G24  can  contain  no  transformation 
of  period  2  or  4  interchanging  the  Qt  (z'=l,  2,  3,  4)  in  pairs.  Also,  G24  can  not 
contain  a  transformation  of  period  4  permuting  the  Qi  (*=1,  2,  3,  4)  in  cyclic 
order  for  its  square  would  be  a  transformation  of  period  2  interchanging  them  by 
pairs.  Hence,  the  G„4  must  contain  a  transformation  Sx  of  period  2  or  4  which 
transforms  the  Q,  (i— 1,  2,  3,  4)  in  the  order  R3=(QaQb)  (Qc)  (Qd)  ;  but 
since  R2R8=(QaQbQcQd)  the  G24  would  then  contain  transformations  permut- 
ing the  Qi(zWl,  2,  3,  4)  in  cyclic  order  which  has  just  been  shown  to  be  impossible. 
Hence  G2880  contains  no  subgroup  Gm  having  a  G4(Q)  as  its  largest  self-conju- 
gate subgroup  of  translations  and  such  that  the  G4(Q)  has  no  system  of  transi- 
tivity which  is  also  a  system  of  transitivity  of  the  Gm. 

Subgroups  containing  a  self-conjugate  G4(P).  Let  Gk  be  a  subgroup  of  G2SP„ 
having  a  G4(P)  as  its  largest  self-conjugate  subgroup  of  translations  and  such  that 
the  G4(P)  has  no  system  of  transitivity  which  is  also  a  system  of  transitivity  of 
the  Gk.  Selecting  the  G4(P)  as  the  group  of  all  transformations  of  the  form 
S:  x'=x-\-a,  y'=y,  and  T:  x'=a1x-\-bly-\-c1,  yf=a2x-\-b2y-\-c2  as  the  general 
transformation  in  the  G2880  we  have  TST"1 :  x/=x-\-a1a,  y'=y-\-aa.,.  Hence  in 
order  that  TST"1  may  belong  to  the  G4(P)  we  must  have  a2=o 
and  if  we  also  have  «,=/  each  translation  in  the  G4(P)  is  self- 
conjugate.  Hence  every  transformation  in  Gk  is  of  the  form  Tt : 
xf=aix-\-b1y-\-cl,  y'=b.,y-{-c2.  The  G4(P)  consists  of  all  transla- 
tions having  /  for  axis  and  4  =(1,0,0)  for  center.  The  G4(P)  has  four  svstems 
of  transitivity,  ^=(0  1  16  14),  /2=3(8  5  18  20),  /3=(2  13  10  9),  /4ss(16  16  12  11) 
and  in  each  system  the  four  points  are  collinear.  From  the  form  of  T,2  it  appears 
that  every  elation  in  Gk  is  of  the  form  E:  x'=x-\-b1y-\-c1,  y'=y  (where  b}  is  not 
zero)  and,  consequently,  has  4  for  center  and  leaves  each  ?i(i«= =1,  2,  3,  4)  invar- 
iant. Hence  there  is  no  Gk  containing  translations  and  elations  only.  From  the 
form  of  T/  it  appears  that  every  transformation  of  type  III  in  Gk  is  of  the  form 
U:  x'=x-j-b1y-\-ci,  v'=y+f2'  where  neither  b1  nor  c2  is  zero.  The  12  trans- 
formations for  which  c2=i  make  the  interchange  {lj2)  (/3/4)  ;  the  12  for  which 
r2=/' make  the  interchange  (IJ^'ilJJ  ;  and  the  12  for  which  c2=r  make  the  inter- 
change (/j/J  (/3/2).    Since  the  Gk  can  not  leave  any  ?t(i— 1,  2,  3,  4)   invariant  it 


COLLINEATION    GrOUI'S   OF    PG(2,22).  J] 

must  either  be  transitive  on  the  /,  or  interchange  them  by  pairs. 
Suppose  Gk  to  make  the  interchange  (/,/.,)  (/.,/4).  Such  a  GK  can 
not  contain  a  homology  having  /  for  axis,  for  the  homology  would  permute  three 
of  the  lines  /j  in  cylic  order.  Hence,  the  Gk  would  be  at  most  (4,  1) 
isomorphic  with  some  group  on  the  line  leaving  a  point  invariant  and  its  possible 
orders  would  be  (cf.  p.  32)  8,  12,  1<>,  24,  48.  Since  Gk  is  transitive  on  8  points  its 
order  must  be  divisible  by  8  and  can  not  be  12.  If  Gk  be  of  order  8  and  make 
the  interchange  (/,/.,)  (/.,/4)  it  must  contain  a  transformation  of  the  form  U, : 
x'=^=x-\-bly-\-ci,  y'=y-|-/.  The  products  of  a  U,  and  the  transformations  in  the 
G4(P)  are  four  transformations  of  the  form  U,  where  Z>,  is  fixed  and  r,  is  any 
mark  of  the  GF(2-).  These  4  transformations  of  type  III  and  the  transforma- 
tions in  the  G4(P)  form  a  group  Gs  which  is  a  Gk.  Hence  there  are  three  such 
subgroups  Gs  (one  for  each  value  of  bx)  interchanging  the  /i(i=l,  2,  'i,  4)  in  the 
order  (/,/.,)  (/:i/4)  and  similarly  3  subgroups  Gs  for  each  of  the  orders  (/,/..) 
(A./;)  and  (lj4)  (IJ.,)-  No  Gk  can  contain  more  than  8  transformations  of  the 
form  U,,  since  the  product  of  \Jl:  x'=x-\-y-\-cl,  v'=y-\-  \J / :  x,=x-\-iyJrfi, 
y'=y+  /,  and  \J  /' :  x'=x+ry+cu  y'=y+ 1,  is  U,  U/  U.":  *'=*+<-,+»,  y'=- 
y-\-r,  which  is  a  translation  not  in  the  G4(P).  The  product  of  any  two  transfor- 
mations of  the  form  U,  and  U,'  is  U1U/=E1:  x'=x-\-i-y-\- 1 ,  y'=y  which  is  an 
elation  having  4  for  center.  The  products  of  E4  and  the  four  transformations  in  the 
G.f(P)  give  all  elations  of  the  form  E,':  x'=x-\-i-y-\-c,  y'—y.  The  product  of 
any  two  elations  of  the  form  E/  is  a  translation  in  the  G4(P).  Also  all  prod- 
ucts of  transformations  of  the  form  E/  with  transformations  of  the  form  \Jl  or 
U,'  are  of  the  form  U,'  or  U,  respectively.  Hence,  the  12  transformations  of 
the  forms  U,,  U/  and  E,'  together  with  the  G4(P)  form  a  Group  G,„  which  is 
a  Gk.  Obviously  there  may  be  two  other  such  groups  G1(5  making  the  interchange 
Ci  OCa  O  and  similarly  •'*  groups  G,„  making  the  interchange  (,/,  l3)(la  /,) 
and  3  groups  GM!  making  the  interchange  (/,  l4)(l-2  /:t).  Also,  since  the  product 
of  an  elation  of  the  form  E:  x'=x-\-by-\-c,  y'=y  and  a  transformation  of  type 
III  of  the  form  U:  x'=x-\-by-\-cu  y'=y-\-c,  is  EU :  x,=x-\-c1-\- bc2-{-c,y'=y-\-c.,, 
which  is  a  translation  not  in  the  G,(  P),  there  is  no  other  type  of  group  of  order 
lfi  which  can  be  a  Gk  interchanging  the  l\  by  pairs. 

A  Gk  of  order  24  or  4S  which  makes  the  interchange  (/,/..)(/.,/,)  must  con- 
tain transformations  of  period  'J.  Every  such  transformation  would  leave  each 
/i  invariant  and  hence  would  be  a  homology  H  having  4  for  center.  Such  a  Gk 
must  also  contain  some  transformation  T  making  the  interchange  (/,  /;>)(/.,  /4). 
But  T  can  not  be  of  type  II,  for  in  that  case  T3  would  be  a  translation  not  in  the 
G4(P),  and  T  can  not  be  of  type  III,  for  in  that  case  HT  would  be  such  a  trans- 
formation of  type  II.  Hence  there  is  no  Gk  of  order  24  or  48  interchanging  the 
l\  by  pairs.. 

No  Gk  can  contain  a  homology  H  having  /  for  axis;  for,  if  P,  be  the  center  of 
H,  Gk  must  contain  some  transformation  T  transforming  P,  to  some  point  P/ 
which  is  not  collinear  with  P,  and  4.  Since  H  can  not  leave  P,'  invariant  HT 
and  TH  transform  P,  to  different  positions  and.  hence,  H  and  T  are  not  commu- 
tative THT',=^-H,  is,  therefore,  a  homology  having  P,  for  center  and  one  of 
the  products  HH,  or  H2H,  is  a  translation  not  in  the  G4(P)  since  it  would  have 
for  center  the  point  where  the  line  P,P/  cut  /.  Accordingly  Gk  is  at  most  (4,  1) 
isomorphic  with  some  group  on  the  line  leaving  a  point  invariant  and  its  possible 
orders  are  8,  12,  16,  24  or  48.  If  Gk  be  transitive  on  the  /,  it  is  transitive  on  all 
points  not  on  /  and  its  order  must,  therefore,  be  divisible  by  1<>.  Consequently  such 
a     transitive     Gk     must     be     of    order     16    or     48.       If     such     a     Gk     be     of 


42  U.  G.   Mitchell:   Geometry  and 

order  16  it  must  contain  two  transformations  of  type  III,  U. : 
;(•'=.*• -f-^iV+^n  y'=y-\-clt  and  U2:  x'=x-\-a2y-\-b2,  y'=y-\-c2  where  c,  and  c2  are 
neither  one  zero  and  are  distinct  from  each  other.  The  product  UjU2  is  of  tihe 
form  U3:  x'=x-\-a3y-\-b3,  y'=y-\-c3i  where  a3  is  different  from  ax  and  a2  and 
c3  is  different  from  ct  and  c2.  Also  a1  must  be  distinct  from  a2,  for  otherwise  U3 
is  a  translation  not  in  the  G4(P).  All  products  U2U3  are  of  the  form  \J1  and 
all  products  1-1,113  are  of  the  form  U,.  Also,  the  square  of  each  Uj(/=1,  2,  3) 
is  a  translation  in  the  G4(P).  Hence  12  transformations,  4  each  of  the  forms  Ut, 
U2,  U3,  such  that  ax,  a.,,  a3  are  no  two  the  same  mark  and  cv  c2,  c3,  are  no  two. 
the  same  mark,  form,  together  with  the  G4(P)  a  group  and  the  only  type  of  group 
G,6  which  is  a  transitive  Gk  of  order  16.  Since  ax  and  cY  may  each  be  chosen  in  3 
different  ways  and  a2  and  c2  may  each  then  be  chosen  in  2  different  ways  thcrfc 
are  36  such  groups  G]0  having  the  given  G4(P)  as  its  largest  self-conjugate  sub- 
group of  translations. 

If  Gk  be  a  transitive  group  of  order  48  it  must  contain  a  transformation  T  of 
period  3.  If  T  be  a  homology  it  must  either  have  some  line  through  4  for  axis  or 
have  4  for  center.  That  T  can  not  be  a  homology,  having  /  for  axis  has  been 
shown  above.  That  T  can  not  be  a  homology  having  any  other  line  through  4  for 
axis  follows  from  the  fact  that  if  S  be  one  of  the  translations  in  the  G4(P)  TS  is 
of  type  II  and  (TS)3  is  a  translation  not  in  the  G4(P).  If  T  be  a  homology 
having  4  for  center  it  leaves  each  /t  ( z==l,  2,  3,  4)  invariant.  Since 
the  G48  must  contain  a  subgroup  G16  which  can  have  no  transforma- 
tion other  than  identity  in  common  with  the  cyclic  G3  generated  by  T,  the 
group  J  G16,  G3  [  is  the  G48  and  leaves  each  /i(z=l,  2,  3,  4)  invariant  unless 
the  G]6  contain  a  transformation  U  of  type  III  having  4  for  center  and  having  for 
axis  a  line  through  P3  some  point  on  /  different  from  Pj  and  P2.  Hence  the  GlS 
would  contain  at  least  24  distinct  homologies  and  since  the  products  of  these  by  U 
give  24  distinct  transformations  of  type  II  the  G48  would  contain  more  than  48 
transformations.     Hence  a  transitive  Gk  of  order  48  can  not  contain  a  homology. 

Since  T  can  not  be  a  homology  it  must  be  of  type  I3  having  for  vertices  of  its 
invariant  triangle  a  point  A  not  on  /  and  two  points,  4  and  some  other  point  Pr 
other  than  4  on  /.  The  cyclic  G3  generated  by  T  together  with  the  G4(P)  gen- 
erates a  G12  consisting  of  all  transformations  of  determinant  unity  leaving  invari- 
ant the  points  4  and  F1  and  the  line  /a  joining  A  to  4.  But  the  G48  must  contain 
a  subgroup  GiC  having  no  transformation  other  than  identity  in  common  with  the 
cyclic  G3  generated  by  T.  Hence  the  G]0  and  the  cyclic  G3  would  generate  a  G48 
leaving  /a  invariant  unless  the  G16  contain  a  transformation  U  of  type  III  inter- 
changing /a  with  some  other  line  /b  through  4.  If  the  other  two  lines  tihrough  4 
be  designated  as  lc  and  /d,  U  makes  the  transformation  U=(/a/b)  (/Jd)  ^n 
these  lines  and  T  makes  the  transformation  T=(/a)  (IbhU)-  Hence  TU= 
(IJJb)(U)  is  of  type  I3  leaving /d  invariant  and  UT=  ( /a/b/d )  ( /c )  is  of  type 
1^  leaving  lc  invariant.  Also,  the  product  of  TU  and  UT  is  a  transformation  of 
type  I3  leaving  the  line  /d  invariant.  Each  of  these  transformations  of  type  I3  generates 
a  cyclic  G3,  which,  taken  with  the  G4(P)  generates  a  G12  containing  8  transfor 
mations  of  type  I3.  These  32  transformations  of  type  I3  are  such  that  each  point 
not  on  /  is  an  invariant  point  of  two  of  them  and  each  point  on  /  (other  than  4) 
is  an  invariant  point  of  8  of  them.  Hence  the  self-conjugate  G1G  can  not  contain 
an  elation  E  ;  for,  if  T'  be  a  transformation  of  type  I3  having  an  invariant  point 
on  the  axis  of  E,  ET'  would  be  a  transformation  of  type  I3  not  among  the  32 
above  named.     The  G16  must,  therefore,  consist  of  12  transformations  of  type  III 


COLLINEATION    GROUPS   OF    PG(2,2S).  [3 

and  the  G4(P).  Taking  the  G10  as  all  transformations  of  the  form  U  :  x'=x-\-ay-\-b> 
y'=y-\-a  and  the  cylic  G.t  generated  hy  a  transformation  of  type  I3  as  all  transfor- 
mations of  the  form  T:  x'=mx,  y'=n  ,  where  mn—i  and  neither  m  nor  n  is 
unity,  the  products  UT:  x'=mx-\-any-\-b,  y'=ny-\-a  and  TU :  x'=mx-\-amy-\- 
</nb,  y'=ny-\-na  are  32  transformations  of  type  I3  as  above  described.  Also,  it  i> 
readily  verified  that  the  product  of  any  two  transformations  of  the  form  U,  UT 
and  TU  is  of  the  form  U,  UT  or  TU.  Hence  they  form  a  group  G48  which  is  a 
transitive  Gk. 

Subgroups  containing  a  self-conjugate  Gs.  There  remains  to  be  determined 
every  subgroup  Gt  of  G2880  which  leaves  fixed  no  point  not  on  /,  has  a  G8  as  its 
largest  self-conjugate  subgroup  of  translations  and  has  no  system  of  transitivity  which 
is  also  a  system  of  transitivity  of  the  G8.  A  G8  of  translations  consists  of  a  G4(P) 
(all  translations  of  which  have  for  center  the  same  point  P  on  /)  and  4  other 
translations  each  having  a  different  center  from  any  other  in  the  G8.  The  Gs 
leaves  invariant  besides  /  and  P  a  pair  of  lines  through  P.  Hence  the  G8  has  two 
systems  of  transitivity  of  8  points  each  and  a  Gt  must  be  transitive  on  the  16  points 
not  on  /.  No  Gt  can  contain  a  homology  H  having  /  for  axis ;  for,  if  T  be  a  trans- 
lation in  the  G8  having  a  point  P'  different  from  P  for  center  H  and  T  are  not 
commutative  (since  HT  and  TH  transform  the  center  of  H  to  different  positions) 
and  HTH"1  is  a  translation  having  P'  for  center  and  not  in  the  G8.  Hence,  a  Gt 
contains  no  transformation  leaving  /  pointwise  invariant  except  the  translations  in 
the  G8  and  is  at  most  (8,  1)  isomorphic  with  some  group  leaving  invariant  a  point 
on  the  line.  Its  possible  orders  are,  therefore,  (cf.  p.  32)  16,  32,  48  and  90.  Tak- 
ing P  as  the  point  4=(/,a,o)  the  G4(P)  in  the  G8  becomes  the  four  translations 
of  the  form  S:  x'=x-\-a,y'=y  and  the  four  other  translations  in  the  G„  may  be 
taken  as  the  four  translations  of  the  form  Sj :  x'=x-\-a,  y'=y-|-/.  Since  a  Gt  can 
contain  no  translation  not  in  the  G8  a  translation  of  the  form  S  must  be  trans- 
formed into  a  translation  of  the  form  S,  and  a  transformation  of  the  form  S,  must 
be  transformed  into  a  translation  of  the  form  St  by  every  transformation  in  Gt.  It 
has  already  appeared  above  that  the  first  of  these  conditions  requires  that  every 
transformation  in  a  Gt  shall  be  of  the  form  Tt:  x'=alx-\-bly-\-cl,  y'=b.,y-\-i... 
Transforming  Sj  through  T1  gives  TS,^"1 :  x'=x-\-aalty'=y-\-b2  and  hence  «,=--l, 
b.1=l  and  every  transformation  in  Gt  is  of  the  form  T2":  x'—x-{-biy-\-cl,  y'=y-\-c... 
If  by=o  or  if  c2=o  T2  is  of  period  2.  Hence,  every  elation  in  a  Gt  is  of  the  form 
E:  x'=x-\-b1y-\-c1,y'=y,  where  bt  is  not  zero.  If  neither  bx  nor  c2  is  zero  T,  is 
of  period  4.  Accordingly  a  Gt  can  contain  only  transformations  of  period  2  or  1 
and  must  be  of  order  16  or  32. 

The  two  systems  of  transitivity  of  the  Gs  are  left  invariant  by  every  elation  of 
the  form  E  and  hence  a  GL  must  contain  a  transformation  U  of  type  III.  Under 
the  G8  of  translations  the  pair  of  lines  y=o,  y«=i  is  one  system  of  transitivity  and  the 
pair  of  lines  y=i,  y=r  (equations  in  non-homogeneous  coordinates)  is  the  other 
system.  Hence  U  must  be  of  the  form  Uj :  xf==x-{-ay-\-b,  y'=y-\-i  or  of  the  form 
U2:  x'=x-\-ay-\-b,  yf==y-\-r-  But  the  product  of  a  translation  of  the  form  S, 
and  a  transformation  of  the  form  \J1  or  U2  is  of  the  form  Us  or  U,,  respectively, 
and  hence  every  Gt  must  contain  transformations  of  both  forms.  A  Gt  of  order 
16  must,  therefore,  contain,  besides  the  QH  of  translations,  4  transformations  of  the 
form  U,  and  4  of  the  form  U2  where  a  has  the  same  value  for  all  of  these  trans- 
formations of  type  III.  Since  every  product  U,U2  and  U2U,  is  a  translation  fn 
the  G.,  these  16  transformations  form  a  Group  G„  which  is  a  (it.  Since  there  are 
3  choices  for  a  there  are  3  such  groups  G,fl  having  the  given  Gs  as  the  largest  sub- 


44 


U.  G.   Mitchell:   Geometry  and 


group  of  translations.  Also,  it  appears  from  the  above  that  every  Gt  must  contain 
at  least  one  such  G1C  as  a  subgroup.  This  G]6  will  be  taken  as  the  subgroup  con- 
sisting of  the  8  translations  in  the  G8,  4  transformations  of  the  form  U/:  x'=x-\- 
y-\-c,  y'=y-\-i  and  the  four  transformations  of  the  form  U/ :  x'=x-)-y-\-c, 
j/=_y-|-r.     For  convenience  this  G16  will  be  referred  to  as  the  group  G'. 

A  Gt  of  order  32  must  contain  some  transformation  T  of  period  2  or  4  not  in 
the  subgroup  of  order  16  taken  as  G'.  If  T  be  of  period  2  it  must  be  of  the  form 
E:  x/=x-\-ayJrb,  y'=y.  The  products  U/E:  x'=xJr{a-\-i)y-[-b-\-cJ 
y'=yJri>  and  U/E:  x'=x-\-.(a-\-i)y-\-b-\-c,  y'=y-\-i~  are  translations  not  in  th^ 
G8  of  translations  if  a=i.  If  et=*i,  the  G32  must  contain  4  transformations  of 
type  III  of  the  form  U/':  x'=x-\-ry-\-m,  y'=y+z,  and  4  of  the  form 
U2":  xf=x-\-ry-\-m,  y'=y-\-i'.  Also,  the  products  of  E  and  the  translations  of 
the  form  S1  introduce  4  transformations  of  the  form  U3:  xf=-x-\riy-\-b,  y'=y-\-  r. 
Thus  are  determined,  besides  the  8  translations  in  the  G8,  4  transformations  of  each 
of  the  forms  E,  U/,  U,"  and  U;i.  These  are  all  of  the  form  U:  x'=x-{-ay-\-b, 
y/=-j.-|_f  where  c=i  or  r  if  o=i  or  r  and  c=o  or  /  if  o  —  0  or  i.  Taking  Ut: 
x'=.x--j-tf1};-|-Z>1,  y'=y-\-Ci  as  a  second  transformation  of  the  form  U  the  product 
is  UUa:  x'=x-\-(a1-\-a)y-\-ac1-{-bi-\-b,  y'=y+f-)-r1.  If  o=o  or  ax=o  or  \i 
0=0-1  it  may  be  verified  by  inspection  that  this  product  is  one  of  the  given  forms. 
For  the  other  possibilities  the  following  table  gives  the  results: 


a 

<*i 

c 

Ci 

Resulting  forms. 

I 

i 

r  or  i 

o  or   i 

U/'  or  Ua" 

I 

V 

v  or  z 

r  or  i 

E  or  U3 

i 

i 

0  or   I 

r  or  i 

U,"  or  Uo" 

i 

P 

o  or  i 

i  or  i'1 

U/  or  U/ 

r 

i 

i  or  r 

t  or  r 

E  or  U3 

r     1 

i      I 

i  or  r 

o  or    / 

U/  or   U./ 

Hence  these  32  transformations  form  a  group  G32  which  is  a  Gt. 
If  a  in  E  had  been  chosen  as  r,  a  G32  of  the  same  type  would  have  been  dete* 
mined.     Hence  there  are  two  groups  G32  having  the  G'  as  a  subgroup. 


SUMMARY. 

1.  In  the  finite  projective  plane  PG(2,2n)  the  diagonal  points  of  a  complete 
quadrangle  are  collinear. 

2.  If  an  outside  point  of  a  conic  be  defined  as  any  point  of  intersection  of  tan- 
gents to  the  conic,  every  conic  in  the  PG(2, 2n)  has  but  one  outside  point  and  all  tan- 
gents to  the  conic  concur  at  that  point.  Through  every  point  other  than  the  outside 
point  there  passes  one  and  but  one  tangent  to  the  conic  and  every  line  through  the  out- 
side point  of  a  conic  is  a  tangent  to  the  conic. 

3.  In  the  PG(2>2n)  six  and  but  six  points  can  be  chosen  such  that  no  three  of 
the  set  are  collinear. 

4.  All  of  the  types  of  projective  collineations  of  the  ordinary  projective  plane 
are  present  in  the  PG(2,2n)  and  the  number  of  such  collineations  in  the  PG(2,22) 
is  60480. 


COLLINEATIOX    GROUPS   OF    PG(2,22).  |.~» 

5.  Every  subgroup  of  the  group  G004ao  of  all  projective  collineations  in  the 
PG(2,22),  except  a  self-conjugate  G20j60  leaves  invariant  a  real  figure  [real  with- 
in the  PG(2f2-)]  or  an  imaginary  triangle. 

().     There  are  8  kinds  of  groups  leaving  invariant  an  imaginary   triangle  and 
their  list  is  given  in  Theorem  11. 

7.  All  configurations  in  tlie  PG(2,22)  and  the  groups  characterizing  them  are 
determined.  These  groups  include  the  finite  groups  of  the  ordinary  projective 
plane.  Consequently,  the  simple  G360,  the  Hessian  (jL,,,.  and  the  simple  G,M  are 
all  subgroups  of  the  Ge0480  and  within  the  PG(2,22)  the  geometric  invariant  of 
each  is  a  real  configuration. 

8  The  subgroups  of  the  G.,MS„  which  leaves  a  line  invariant  are  chiefly  (1,1) 
or  (3,1)  isomorphic  with  groups  on  the  line,  but  certain  groups  of  higher  isomor- 
phism are  present  and  are  determined. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

am    INITIAL    FlNE~OF    25     CENTS 

A      .     ^ASSESSED   FOR   FA.LURE  TO   RETURN 

E?  rNoR  to  EJ0oo50oCNENtThe° Seventh   o*v 

OVERDUE. 


OCT   24   1932 


17  1932 


0#294$P 


\2fc' 


3  Apr '49  FA 


NOV     3  1936 
OCT  7 


i2May'49JS 


AV 


Auf(W&*r- 


LU  21-50?n-8,'3-J 


2880 

61 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


>,ir  *,fy 


